100 Prisoners

Recently, Pat wrote an article discussing the need for luck, and deliberately contradicted a prior article in which he said luck doesn’t exist. In short, he says players don’t need luck, yet they do. If you take away the smoke and mirrors, there is no real contradiction. What he describes boils down to improbability vs. variance. They are two different measurements that could perhaps both be thought of as luck, in the sense that we have a minimal amount of control over them. I will explain both concepts in this article.

Contradictions

Before I begin, I would like to highlight my favorite excerpt from his newer article.

"I love to contradict myself. It’s a great way to learn about something and it helps you avoid one of the most common biases; arguing to win the argument instead of arguing to find the right answer."

I want to emphasize his introduction. When a person presents a point, it is clear when his only goal is to show that he is right vs. when his goal is to gather information. This will often reflect in the way he speaks to the person he disagrees with, which includes his tone, volume, facial expression, and choice of words.

When your intent going into any sort of discussion, whether online or in person, is to arrive at a better answer than the one you have, you experience numerous benefits. Your likelihood of finding a better answer increases. You feel no pull to raise your voice or argue out of anger. You will not see those who contradict you as enemies, rivals, or opposition. As a result, accusations and insults will be absent from discussion.

When one activity you do changes the way you think about everything else outside of that activity, a beautiful moment occurs. Deliberating a point has been one of those things for me. Competition has changed the way I see human discussion, and Pat’s intro succinctly captures how Yugioh has effected change in that aspect in my life.

That’s enough personal musing for now. On to today’s topic!

I’m going to have to contradict Pat’s point. While his observation about intentional self-contradiction is magnificent, I don’t think it applies to the examples he uses in the article. His article describes two different observations that both can be labeled as luck and lumping them together. This leads to an apparent contradiction. However, just as there is no spoon, there is no contradiction.

No spoon

Outcomes Over Time

The first kind of luck, the one that he and I don’t believe in, is what the community often calls sacking, bricking, and so on. They are real in one sense because when evaluated only within the context of one game, they occur as the product of chance. However, across a lifetime of competition, these events can be taken together as a set of improbable events occurring at the rate they ought to be occurring at. Because their rate of occurrence matches their theoretical probability, we don’t like to call it luck.

For instance, if one play is punished by Book of Moon and the other is punished equally as much by Mirror Force, I should assume that the set is Book, unless I have a read. This is because (in Goats) you’re likely to have twice as many Books as Mirror Forces. I’m going to play as if it’s Book every time I’m in that situation. Eventually, Mirror Force will get me, and when it happens, I’ll accept it. That won’t change the fact that I should and will play that same situation out like it’s Book each time, everything else being equal.

The reason we don’t call the Mirror Force blowout “luck” is because it statistically ought to happen. The more games I play, the more times I will get hit by that improbable outcome. And that’s fine because as long as I play correctly, I will get hit by that outcome fewer times than people who do not play correctly. If that outcome should occur 25% of the time but it has happened to me in half of all games so far, that still isn’t bad luck. I just need to play more games, and eventually the results will self-adjust, and the outcome will reach a cumulative frequency at or near its expected rate. Human error such as nonrandom shuffling may alter that slightly, but not by any significant amount.

Variance

The second phenomenon Pat labeled luck is, for a statistician, variance. If I tell you that a set of five numbers has a mean of 50, you would need more information to form a conclusion about the variance. That set of numbers could be 49, 50, 50, 50, 51. It could also be 0, 29, 60, 61, 100. Both sets have a mean of 50, but the second set has more variance.

The following is a thought experiment would be really hard to create in reality, but suppose you have a group of students about to take a test, and suppose you could accurately assess that each student has exactly 75% of the content knowledge of whatever the test is testing them on. If the group of students is small, their mean score might not be 75% at first. The larger the group of students, the more their mean score should converge toward 75%. Also, if we set up our experiment right, the range of scores shouldn’t be too far apart. That is, the difference between the lowest score and the highest score should not be a large one. If it is large, then we may have to dig around for the source of our static noise.

Ok, so why is variance good if the means don’t change? Let’s say I win 70% of my games and this is a law I cannot subvert. What if I could manipulate my results such that I can choose which 70% games I win? Obviously I’m going to pick the ones that matter the most. When it’s time to pay my dues and take my 30% losses, I’m going to distribute all of them to times after I’m already X-3. An uneven distribution of win/loss makes the wins more rewarding and the losses inconsequential. Obviously, no one can do this in real life. Rearranging our results while keeping the net wins the same would be a magical feat. However, there are small decisions we can make to slightly encourage variance in our favor.

Rolls

Pat brought up the example of dice rolling. Regular game dice, the round-edged ones that you buy at locals, cannot roll all six numbers in even distribution. Casino dice can. They have sharp corners. You can get them online for really cheap. Nobody uses those, so when we talk about dice, we’re using regular dice. Over time, you’ll win 50% of your rolls and lose 50% of your rolls. Differences among dice and among rolling method will all cancel out in the long run. I can explain why but readers don’t seem to like it when I stray from the topic of Yugioh, haha.

Another way to determine who goes first is to use the Rock, Paper, Scissors cards (NOT RPS, the game. Know the difference). You shuffle the three cards facedown and pick one. It doesn’t matter if your opponent picks from one set of 3 and you pick from another, or if you both use the same set of 3. It doesn’t matter who picks first either. It doesn’t matter if your shuffling is imperfect. It doesn’t matter if your opponent always likes to pick the middle card of the three. All of these potential flaws are automatically eliminated with this method, provided your cards are sleeved and are indistinguishable from one another. This is a very elegant little game whose rules inherently negate the flaws of other types of “random” one vs. one games. Thus, the method will yield 50/50 results.

The difference is that the RPS cards will produce little variance compared to hobby shop dice, even though both produce a longitudinal result of 50/50. The increased variance of rolling dice is more attractive to Pat because he argues that going first exactly half of all game 1s throughout an event is not enough to “get there.” It is better to win a disproportionate number of dice rolls at the risk of losing a disproportionate number of them. Referring back to my magical 70% win scenario, it should be apparent why it’s better.

The difference between going X-3, X-4, and X-5 doesn’t matter. Why would you want a loss when you’re X-1 if you could take that loss when you’re already X-3? It’s the same with rolling dice. Winning the roll round after round has cumulatively stacking benefits during the same event. However, losing the roll round after round does not have cumulatively stacking consequences at the same rate. The consequences plateau in the same way that they do for losses in the magical scenario from earlier. I’ll put it another way. Winning the (n+1)th die roll after winning (n) rolls produces benefits greater than losing the (n+1)th die roll after losing (n) rolls produces consequences.

Decks

Another example of choosing variance over uniformity can be seen in Billy’s deck choices. Particularly before top cut involved Battle Pack draft, Billy opted for decks meant to run through a top cut undefeated, rather than decks meant to reach top cut. That’s because he’d rather win and lower his chances of topping than top but give himself a low chance of winning. Billy’s results generally reflect his approach. He tends to run deep in top cut playoffs, but he also bombs more than he needs to in Swiss.

Would you rather have a 5% shot at winning any given event and a 20% shot at topping, or a 1% shot at winning and a 50% shot at topping? Don’t get hung up over my made-up numbers, but rather consider the general principle behind the question. Which choice appeals to you more will determine your outlook on whether you want to pursue long-term variance outside of individual games or minimize it.

I think Qliphort is a great illustration of low variance philosophy from the previous format, and Geargia from last year’s WCQ format. Both decks have the stability to reach top cut. Going X-1-1, X-2, X-2-1, depending on what event we’re talking about, is a reasonable expectation for these kinds of decks. However, running through the last 5 rounds of playoffs undefeated? Not likely. “Not bricking” is a nice approach to Swiss, but an explosion factor is essential to playoffs. For that reason, the chances of Geargia winning last year’s WCQ, despite its status as a leading deck earlier in the same format, was essentially nil.

My most extreme example is Evilswarm. Evilswarm is what Sirlin would call a “kingmaker.” Its presence changes the recipient of the crown, but it itself does not receive the crown, except once in a blue moon.

Devil's Advocate

When Pat says luck doesn’t exist or doesn’t matter, he means that averages will establish over time. When he says that luck is needed, he means that it’s nice to have high variance, even if the means come out the same. Pat is a good devil’s advocate, but on this matter, he was being a bit dramatic by saying he was contradicting himself. That’s the end of this week’s article. I included something just for fun below, but it is not essential.

The Impossible Bet

The reason I wrote this article, besides calling out Pat for being a drama queen (just kidding, buddy), is because Pat’s perspective on variance reminded me of a fun riddle. It was called the Impossible Bet when I first saw it, though it is known more classically as 100 Prisoners.

Here is one way it was presented in Analytic Combinatorics.

The director of a prison offers 100 prisoners on death row, which are numbered from 1 to 100, a last chance. In a room there is a cupboard with 100 drawers. The director puts in each drawer the number of exactly one prisoner in random order and closes the drawers afterwards. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. If during this search every prisoner finds his number in one of the drawers, all prisoners are pardoned. If just one prisoner does not find his number, all prisoners have to die. Before the first prisoner enters the room, the prisoners may discuss their strategy, afterwards no communication of any means is possible. What is the best strategy for the prisoners?

And a visual presentation: https://www.youtube.com/watch?v=eivGlBKlK6M

If each prisoner simply opens any 50 boxes of his random choosing, hitting the right number becomes a coin flip. With 100 prisoners, that’s trying to flip heads 100 times in a row. Their chances of survival go out past 30 decimal places. Essentially, their chances are zero, and they’re going to die.

You can look up the answer, but I think it would be really gratifying to figure it out independently first. Finding the answer doesn’t require math. Proving that the answer is the answer requires math, but proofs are for nerds and we don’t need to do them.

The solution to the problem yields a 31.1% chance of winning, several hundred octillion times better than opening boxes at random. The solution is also the most elegant demonstration of lumping outcomes I’ve seen. Think of it as a metaphor for manipulated variance, with 50 boxes representing the unchanging mean.

Similarly, variance is used advantageously in Yugioh. Again, this doesn’t have to do with individual games (like the chances of drawing individual cards), but with matches and tournaments over time. It obviously won’t multiply your tops by octillions (or even by 2, for that matter), but slight advantages are still advantages.

Until next time,

Play Devil’s Advocate or Go Home!

Johnny Li

Johnny Li

Houston, TX
Johnny Li

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