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I am currently in the process of refactoring a card game that I learned a couple days ago, the French Canadian Mitaines, or Mittens. It’s an unusual game that I have known about and been mildly fascinated with for a while, almost exclusively because the three combinations that players are competing to build are called mittens, gloves, and socks. A friend and I learned it Wednesday night and played a couple hands, and the mechanisms are actually interesting and relatively novel; basically you play to a central discard pile and try to play your mittens, gloves and socks without being interrupted by the other player. But it’s sub-optimal; as of right now mittens, which are pairs, are quite common, but gloves and socks, which are three and four of a kind, are quite rare.
So I decided there was some room for improvement. The two obvious paths to drawing more combinations without seriously mucking with the rules are to reduce the number of ranks in the deck and to increase the size of the hand. But I was interested in being able to model exactly how the probabilities might be altered by adjusting the various quantities in the game, which meant I needed to model how mittens, gloves and socks arise—which meant I needed to relearn how to use binomial coefficients.
Which I did! I spent a pleasant day yesterday fiddling and I believe I’ve come up with a useful model of how things arise in this game, which means I can now change any variable I like and see how it would be reflected in the game.
You can see my handiwork.
First I modeled the probability of each combination arising—I was very confused that under certain configurations my numbers were easily reaching above 1, until I realized that I wasn’t measuring the likelihood that one would draw, say, a pair, but rather measuring the average number of pairs per hand. Then I established a consistent ratio of all the outcomes’ likelihoods against the least common basic outcome—the sock, or four of a kind—and finally established an inverse function of the ratios to set more rational scores for each outcome than the frankly haphazard and imbalanced scoring that was delivered by the rules we read.
Now I have a task that’s just as challenging: I need to fix a deck configuration for both 2- and 3-player configurations. The standard rules, you could say, might as well be optimized for this task (at the expense of all others, as my judgments above would indicate): A four card flop at the outset out of a full deck of 52 cards means that two players can play with six-card hands, and three players can play with eight-card hands—a very simple adjustment. If you want to reduce deck size and/or increase hand size (preferably and) in order to optimize for higher-scoring combinations, it becomes much more delicate to try to balance the three- and two- player experience. One would like to be able to adjust the number of players without too much fiddling with gameplay; I am resigned that there will have to be more fiddling than in the standard rules.
One unexpected but pretty repeatable result from this work: increasing the hand size is sizably more effective at reducing the ratio between mittens and gloves than reducing the deck size. I was fairly convinced of the opposite.
So if we prioritize a hand size of 8, then we can propose the following tentative setup:
The difference in ratios is a little high for my comfort. I would like to try adjusting the flop size.
Actually, I’m not sure if that is gonna make a difference. I think the only real options here if we want to keep hand size at 8 is to have 32 cards going to two players in two deals, or 24 cards going to three players in one deal. One of the side effects of such a large hand size is that it’s quite inflexible for determining deck size.