Monday, May 24, 2010
Play to Win: The Mathematics of Good Players

More extracts from my Bandai postings that I find useful:

A Swiss + cut event is designed to largely negate the effects of luck (due to the fact that it's possible to make top cut with 1-2 losses). In a deck with 11 0 drops and 3 Tobis (Tobi counts as reducing deck size by 1 - he actually should contribute more than that to probability but we're being conservative here), there is approximately a 18.1% chance of not drawing a 0 drop on a hand of 6, and a 24.6% chance of not drawing a 0 drop on a hand of 5. Assuming that the probability of winning with a hand of 4 is 0, that means there's only a 4.4% chance of drawing a hand that cannot win, no matter the skill level.

And let's assume that the deck is poorly constructed enough that 15.6% of the time, the hand will be unplayable or the cards drawn will do nothing. That means 20% of the time, a player will lose and be unable to control it at all.

6-2 with good breakers usually makes T8 at a 8 round event, 6-1-1 and 6-0-2 always will.

Even if you fail to win in 25% of your matches, you still make top cut.

Now take that earlier 20% - if a player is skilled enough, they'll be favored to win the rest of the 80% of the matches because they are better than their opponents.

But here's the kicker: bad luck doesn't just apply to yourself - bad luck also applies to your opponents. Therefore they also have that 20% chance of flat out losing a match due to a bad draw.

Applying the proper probabilities, here's the breakdown of a match between a good player and a bad player.

64%: Balanced game - good player wins (80% * 80%)
4%: Crappy draws on both ends - good player wins (20% * 20%)
16%: Bad player has a crappy draw - good player wins
16% Good player has a crappy draw - bad player wins

The good player should be expected to win 84% of the time, not just 80%.

Of course, the actual good player win percentage should be even higher. What separates a good deckbuilder from a bad deckbuilder is that the good deckbuilder reduces that 15.6 "unwinnable" percentage by tightening up the variance of the deck. What separates a great deckbuilder from a good deckbuilder is that the great deckbuilder will skew the draw distribution of the deck in order to create auto-win draws where the opponent can simply do nothing, no matter how they draw.

Is it possible for a bad player to be super lucky and make top cut, or a good player be super unlucky and miss top cut? Of course - it's possible, but improbable. Over time, all that shading of probability leads to a situation where good players should be able to top cut consistently due to the Swiss + cut format.


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Blogger Canti said...
I enjoyed your post. Too bad it went far over the intended target's head.

Blogger Ross134 said...
I don't believe this is for those who are at the lower end of the game. More for those of us who usually can place well but worry sometimes about top cut with records of x-1, x-1-1, or x-2

Know the percentages just reaffirms the playability of certain decks and certain players.

Blogger Zounder said...
The mathematics are sound, but the flaw in the logic comes from assuming there's two types of hands: unwinnable and non-unwinnable (winnable). Using this algorithm for deciding who wins a match, if Player A and B both have non-unwinnable (winnable) hands, then whoever is more skilled wins. This is false, because there are degrees of non-unwinnable hands, say, on a scale of 1-10, 1 being unwinnable, 10 being autowin. I think, in a matchup of 2 vs. 9, even if 2 is more skilled, he will still lose unless he is way more skilled.

Therefore, the numbers reached in this post can really conclude nothing about chances of winning. In order to do so, one must consider the probability of winning with each degree of winnable hands, and ALSO the difference in skill in the opponent.

I find, in practice, that players with minimal skill differences often win/lose based on a degree difference of winnable hands. This raises the luck factor quite a bit.

All that being said, I feel like it's still obvious that skilled players consistently top. The luck starts to become apparent when it's one-and-done (top cut).

Blogger Josh said...
I didn't want to make it too complicated - Tom Yu and I already had a discussion about attempting to rank hands in terms of playability and found that's pretty much a futile exercise. In order to draw a conclusion about how exactly Swiss events influence the composition of the top cut, it's necessary to simplify the problem and engage in a few assumptions.

The numbers don't really mean anything - the first thing that should have tipped that off is that I just assumed 15.6% "blah" hands that don't really do anything - the actual number can only be found through repeatedly playing games and seeing which hands actually DO do nothing (playtesting!). I chose 15.6% because not only does it add up with 4.4% to equal a nice even 20%, but also because it's far greater than the number of hands (of both 5 and 6) that actually do suck with a good deck (imo).

The second assumption that makes the numbers meaningless, as you mentioned, is that with hands in equal categories, the good player will always beat the bad player. The reason I made this assumption was that to my knowledge, the difference between the average Naruto player and one of the top-tier players is huge. When Joe Blow plays against Joe Colon, while Joe C may not win 100% of the matches with playable hands, he'll win the vast majority of them.

But to figure out the exact percentage of those matches he wins - I don't think I can do that. To simplify, it's easier to assume that that Joe C will win all of them.

The purpose of the exercise isn't the arbitrary number that I arrived to at the end, but rather the process of going through the exercise. It demonstrates why, exactly, top players can consistently top cut even though luck does play a role in the game.

Still would be nice to find a way to actually calculate this sort of stuff though - maybe assign each hand + draw an expected win % and then skill serves as a multiplier? Ehhh... spending that time playtesting would probably be more conducive to raising EV.

Blogger Zounder said...
I never questioned your assumptions or means of simplifying (you're correct in saying that it would near impossible to get an accurate percentage), my point was just that simplifying makes the results skewed to the point where it's not really accurate.

Blogger Josh said...
The "exact number" result was never supposed to be accurate - just the concept that the Swiss format favors the skilled player. While the percentages were arbitrary, they were also best estimates rather than just random.

I feel like either we're saying the same thing or I'm just missing something somewhere.

Blogger Zounder said...
By not really accurate, I mean inaccurate to the point where it's pointless to mention. It would have been just as sufficient to just say "Skilled players should top" because the numbers don't support that claim since they are too simplified.

In other words, it's really impossible to quantify (by any means simple or complex) stuff like this.

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