Saturday, October 5, 2013

Using Sampling to Make the Best Yugioh Deck Ever!

So you have this nifty idea for a deck. Great! You gather the requisite cards, shuffle them up, and deal out six cards.  Hmm, not bad.  Shuffle again, six more cards, and again … not bad.  Maybe you’re onto to something.  So you go through this drill five more times and then on the eighth time you draw garbage.  At that point, you can pitch the deck and go back to your Dark Worlds, conclude that the deck produces dead hands 12.5% of the time, or have the statistical savvy to know that eight hands is not enough to fully assess the deck’s potential.  Hopefully, you believe the latter.

There are 3,838,380 combinations of six cards in a 40 card deck – well, actually less since there are duplicate cards, but you get the point.  To determine how often your new deck will produce dead hands, you have to take a sample. It’s like political polling where you gather opinions from a small number of people to make statements about the whole group.  In this case, you want to look at a small number of random hands and assess them as “Live” or “Dead”.  The value of this exercise depends on (1) the number of hands you look at and (2) your ability to assess “Live” or “Dead”.  In fact, the latter is probably more important.  The reason that most of us struggle to top with a net deck by Billy Brake is that he sees plays where the rest of us see dead.  If I knew the secret of that skill, you’d be reading about my latest YCS top.  Instead, I can give you the answer to the sample size question. 

I won’t go into all of the mathematics supporting these ideas because that would drive away the small number of readers that I have.  Instead I will try to provide a handful of principles and some useful numbers.  

You could choose to analyze your deck by asking the following question: “How many dead hands are in this deck?”  This question sees the percentage of dead hands as an inherent quality of the deck.  It is measurable like measuring blood pressure in the residents of a group home.  To determine how many of the 3 to 4 million hands you need to look at, you must pick a confidence interval and a confidence level.  The interval is the plus/minus figure that often accompanies polls.  If you choose a confidence interval of 5%, you’re saying that your sample could have 12% dead hands but the deck could have between 7% and 17% dead hands.  The confidence level is the percentage of times you will get the right answer if you sampled the deck multiple times.  For example, if you and nine of your friends sampled the deck and measured the number of dead hands, one would get the wrong answer if you were using a 90% confidence interval.   

So how many hands would you have to look at in order to get an idea of the number of dead hands?  For a 5% interval and a 95% confidence level, you would have to look at 384 hands.  You could back off on your assumptions and use a 10% interval with a 90% level.  In that case, you would need to look at 68 hands.  The main conclusion that I draw from this analysis is that players probably make decisions about their decks with insufficient information (i.e. too few samples).  

Common sense will tell you that you can get a good idea of how good or bad a deck is long before you look at the 300th sample hand.  The reason is that you are intuitively comparing the new deck with your experience and expectations with the old deck.  You are, in a sense, performing a statistical test similar to a Chi Square test.  Like many statistical tests, this one helps answer the question Is my nifty deck different than my old deck?  The bigger the difference (i.e. the more dead hands your nifty deck produces), the fewer hands you will need to look at.  

With that in mind, consider the following table that compares your new deck with a deck that produces live hands 90% of the time.  The first column is the percentage of live hands your new deck produces and the second column is the number of hands you need to look at to determine its inferiority.   If your deck draws good hands only 50% of the time, you will be able to conclude the deck is worse than 90% in 6 hands (line 1).  If your deck produces live hands 70% of the time, you’ll need to look at 24 hands to determine if the deck is worse than your old deck.  If your new deck is 5% worse than your old deck, you’ll need to look at 370 hands.  
% of Live Hands Total # Live hands Live hands
in the Nifty Deck of Hands in the Old Deck in the Nifty Deck
50% 6 5 3
55% 8 7 4
60% 11 10 7
65% 15 14 10
70% 24 22 17
75% 42 38 32
80% 93 84 74
85% 370 333 315

I hope this sampling discussion helps you understand the most common Yugioh complaint I've heard: I didn’t draw $#!^.  No, actually your deck draws much more $#!^ than you realize. You simply haven’t looked at enough hands to know.

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