Wednesday, August 3, 2011

Lies, Damn Lies, and Chutes and Ladders

Okay, parents, you've all been there. Bedtime for the little one is 8:00, and at 7:50 comes "Just one more game!"  My son was really into Chutes and Ladders a few years ago, and for some reason this was the game of choice for sleep procrastination. But when the clock is ticking, the dice seem to conspire to make the game go on... and on... and on...

So, how many turns should one expect a game to last? Theoretically, this game could go on forever, but statistically one should expect a finite end nearly all of the time. Not surprisingly, people have thought about this problem... quite a bit in fact. (See also discussion of the mathematics of the game in the Wikipedia article linked above.)  My fellow geeks seem to agree with my own analysis that the mean number of turns that a player should expect is about 39.5, and the median is about 33.

But not so fast!  All of the calculations that I've seen assume (rightly) that a player of Chutes and Ladders doesn't affect the actions of the other players at all.  Thus, the computer programs and mathematical equations just tend to simulate a single player, from his or her start in space 1 to the end in space 100. But for two or more players, only ONE of them will follow this single-player simulation all the way to the bitter end.  The others will stop when the winning player wins, and this skews the distribution!  In other words, when you have more players, you have more chances of ending the game with a low number of turns!

So I fired up the old random number generators, and wrote some code in IDL to simulate a few hundred thousand games... Here are the histograms for different numbers of players:
The y-axis shows the relative probability of winning the game in the number of turns shown on the x-axis. The mean number of turns for a 1 player game is 39.7, for 2 players it's 26.5, for 3 players it's 21.8, and for 4 players it's 19.3. Another interesting change is the eating away of the one-player "long tail" to give multiple players a much more compact distribution. I'm sure that one could have modeled these Poisson-like distributions without crunching through millions of simulated moves, but computers just make doing the latter so easy! :-)

I also played around with a common house rule that we implemented when our son was very young: once you get into the final row (spaces 91 to 100), you're home free and the chutes don't count. For the one-player game, using this rule reduces the mean number of turns from 39.7 to 31.6, but the median goes down only from 33 to 28.  Not a huge time-saver, I suppose, but it may reduce player crankiness!

(Of course, adding more players isn't necessarily all peaches and cream!  More players may also mean more arguments about who gets which counter, who spins the spinner properly, who goes next, etc... so caveat emptor!)

Wow... when I started thinking about doing a post on Chutes and Ladders, I was intending to talk mainly about its origins as an ancient Indian teaching tool for the concepts of karma and reincarnation, and maybe about Deepak Chopra's recent morphing of the idea into a weird video game. As a kid, I think I may have encountered its Tibetan variety even before I'd ever played the Hasbro version!  :-)

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