A blog of tips and recommendations for anyone interested in learning or teaching mathematics.
Wednesday, November 18, 2015
Perfecting the imperfect [repost]
[disclaimer -- I've only field tested the first of these, so I can't guarantee that all of the variations will play smoothly. On the bright side, there ought to be plenty of room for improvement. As with all discussions of game variants, you should probably assume that countless people have already come up with any idea presented here.]
When the subject of perfect information games comes up, you probably think of chess, checkers, go, possibly Othello/Reversi and, if you're really into board games, something obscure like Agon. When you think of games of imperfect information, the first things that come to mind are probably probably card games like poker or a board game with dice-determined moves like backgammon and, if you're of a nostalgic bent, dominoes.
We can always make a perfect game imperfect by adding a random element or some other form of hidden information. In the chess variant Kriegspiel, you don't know where your opponent's pieces are until you bump into them. The game was originally played with three boards and a referee but the advent of personal computing has greatly simplified the process.
For a less elaborate version of imperfect chess, try adding a die-roll condition to certain moves. For example, if you attempt to capture and roll a four or better, the capture is allowed, if you roll a two or a three, you return the pieces to were they were before the capture (in essence losing a turn) and if you roll a one, you lose the attacking piece. Even a fairly simple variant such as this can raise interesting strategic questions.
But what about going the other way? Can we modify the rules of familiar games of chance so that they become games of perfect information? As far as I can tell the answer is yes, usually by making them games of resource allocation.
I first tried playing around with perfecting games because I'd started playing dominoes with a bluesman friend of mine (which is a bit like playing cards with a man named Doc). In an attempt to level the odds, I suggested playing the game with all the dominoes face up. We would take turns picking the dominoes we wanted until all were selected then would play the game using the regular rules. (We didn't bother with scoring -- whoever went out first won -- but if you want a more traditional system of scoring, you'd probably want to base it on the number of dominoes left in the loser's hand)
I learned two things from this experiment: first, a bluesman can beat you at dominoes no matter how you jigger the rules; and second, dominoes with perfect information plays a great deal like the standard version.
Sadly dominoes is not played as widely as it once was but you can try something similar with dice games like backgammon. Here's one version.
Print the following repeatedly on a sheet of paper:
Each player gets as many sheets as needed. When it's your turn you choose a number, cross it out of the inverted pyramid then move your piece that many spaces. Once you've crossed out a number you can't use it again until you've crossed out all of the other numbers in the pyramid. Obviously this means you'll want to avoid situations like having a large number of pieces two or three spaces from home.
If and when you cross off all of the numbers in one pyramid you start on the next. There's no limit to the number of pyramids you can go through. Other than that the rules are basically the same as those of regular backgammon except for a couple of modifications:
You can't land on the penultimate triangle (you'd need a one to get home and there are no ones in this variant);
If all your possible moves are blocked, you get to cross off two numbers instead of one (this discourages overly defensive play).
I haven't had a chance to field test this one, but it should be playable and serve as at least a starting point (let me know if you come up with something better). The same inverted pyramid sheet should be suitable for other dice based board games like parcheesi and maybe even Monopoly (though I'd have to give that one some thought).
I had meant to close with a perfected variant of poker but working out the rules is taking a bit longer than I expected. Maybe next week.
In the meantime, any ideas, improvement, additions?
Originally posted in West Coast Stat Views
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