A Clear Plea for Intervention
I can’t just play games. As much as I love chaos, I love full and complete understanding of the math behind systems even more. I end up thinking like a powergamer most of the time just because of my incessant drive to break whatever game I happen to be playing. I don’t like challenges except insofar as I like absolutely destroying them with creative thinking and/or logical deduction. Games like Trouble or Sorry have absolutely no appeal to me. Games like Final Fantasy Tactics or Neverwinter Nights can take over my life.
The current object of my obsession is the Marvel Trading Card Game for the PSP and, more generally, the entire Upperdeck VS System. After getting my ass kicked repeatedly by computer opponents, my geek pride was bruised. I started digging deep into strategies of powergamers in a quest to find the distribution of character cards to plot twist cards and nearly got overwhelmed by the sheer variety of decks and strategies possible in the game. What I needed was a way to quickly try out scenarios, so that I could see the difference in the overall curve that adding or subtracting a card would cause.
I reached for Excel. I understand that, as a Unix geek, I should probably be whipping up a script in Perl or Ruby or something, but honestly, Excel is my happy place when it comes to quickly banging out mathematical formulas. In fact, when I’m needing to work out an algorithm for heavy calculations, I’ll usually work out the math in Excel before I start coding in the appropriate language. I did some research into the calculations I would need to work out for the problem at hand and found out that I was looking at a hypergeometric distribution. Conveniently, Excel has a function just for this sort of thing. To calculate out the probability that I’ll draw the card I need on an appropriate turn, I just do something like this:
=1-(HYPGEOMDIST(0,6,6,60))
In essence, I’m calculating out the probability that I won’t draw a certain card and just subtracting that from 1 to find out the probability that I will draw at least one of that card. In the above formula:
- 0 is specifying that the number of cards that I want to test for. (Remember that I’ll just be subtracting this probability from one so that I’ll get the opposite probability.)
- The first 6 is the number of cards I have drawn at this point. Since I’m calculating out round 1 without taking a mulligan, I will have only drawn 6 cards by this turn. If I invoked the mulligan rule, I will have drawn 10.
- The second 6 is number of a certain type of card that I have in the deck. (In this case, 6 2-drops.)
- 60 is the number of cards in my deck.
I basically set up a grid of these calculations, allowing me to calculate the following probabilities:
| Card Cost | Quantity | Chance of Drawing by Correct Turn |
|---|---|---|
| 1 | 0 | - |
| 2 | 6 | 75% (mulligan) |
| 3 | 8 | 79% |
| 4 | 7 | 81% |
| 5 | 6 | 81% |
| 6 | 5 | 80% |
| 7 | 4 | 81% |
With this distribution, I have about a 75-80% chance of hitting each of my appropriate drops (with the exception of usually worthless 1-drops). That’s a chance I can definitely live with. I’m not adding in any 8-drops because I fully expect games to be over before turn 8. My final deck will consist of 36 character cards, 4 locations, and 20 plot twists. Now, I’ll have to see how playing such a deck shakes out, but math tells me that it would have to be more successful that my current decks (which hover at around 50-60% rates).
See what I mean? I did all this for a game that I’m playing for fun. And the part that will seem truly odd to some of you out there is that these calculations are a major part of the fun for me. If I can’t ponder questions like this, then I’m much less likely to be interested in a game.
Now I’ve just got to win enough games so that I can afford to buy enough cards to fill out this distribution…Then comes the fun of real world playtesting!
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