I'm currently in the process of reconfiguring a card game that I learned a couple days ago, the French Canadian Mitaines, or Mittens. It’s an unusual game that I've 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.
Three of a kinds and four of a kinds are naturally going to be considerably rarer than pairs in any card game, but there are certain things we can do to increase the likelihood of their appearance.
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 on Thursday 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.
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.
We can see that threes and fours of a kind are now not only more plentiful, but more importantly they're in a less dramatic relation to pairs—the event ratios don't ramp up as quickly. Thus the scores for each combination can generally be higher, as pairs are not so dramatically more likely than every other hand.
(My thanks, incidentally, to Greg Rae for holding my hand through the process of actually writing up the functions describing combination frequency.)
Now I have a task that’s just as challenging: I need to fix deck configurations for both 2- and 3-player games. The standard rules 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; that is, preferably you'd like to make as few adjustments as possible in terms of flop size, deck size, hand size, et al when moving between game types. 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 seems to be considerably more effective at reducing the ratio between mittens and gloves than reducing the deck size. I had been fairly convinced of the opposite before I started with this.
So if we prioritize a hand size of 8, then we can propose the following tentative setup. The socks:gloves:mittens ratio can be a vague measure of the flavor of the gameplay; I'd like the numbers to be somewhat similar so that the game plays itself out in somewhat similar terms, and so the point values of each combination (which surely cannot change from game mode to game mode, lest my players mutiny and go back to Euchre) will be reasonably appropriate.
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's 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.
A few notes on minor features of the models:
Actually, I do wonder if I am underestimating the influence of adding on combinations here. I wonder if it's expected that the vast majority of threes and fours should be made by adding on to somebody else's pair.
Playtesting has been informative. I'm happy to say that the proposed hand and deck arrangements provide for much livelier play.
I've also been experimenting with different scoring schemes; that's where the remaining work lies. One quality that I failed to fully appreciate was the simplicity of the original system—not just in terms of the arithmetic needed to sum up one's score (everything being 10, 50, 150, 250, etc.) but also in the mnemonic elegance of how the scores are represented by captured cards.
In the original system there are really only three scoring tokens, that is, captured cards used to keep score and add up at the end of the game: face-up cards are worth ten points, face-down cards are worth 100 points, and 10♦ and Jacks are worth fifty points. Every scoring unit—100 for a glove, 200 for a sock, 150 for a glove of Jacks, etc.—can be composed of those three items.
Whatever the relative fairness or proportionality of those specific amounts, the value of a system which only has three tokens to mix and match can't be overstated. In our playtesting, the adjusted-scoring games (notwithstanding whatever degree they need further adjustment, for reasons stated above) weren't nearly as fun or smooth as the ones with the classic scoring simply because they required so much more effort and bother.
It's clear, if I do want to adjust the scores for fairness, that a similar system—three scoring tokens at max—will have to stay.