Ridere, ludere, hoc est vivere.

Thursday, November 8, 2012

Running the numbers in East India Company

I've mentioned several times that I'm worried about game length in "East India Company."  Each turn has a "New Colony Phase" in which a tile is drawn from a bag and added to one of the seven colonies on the board.  There are 21 tiles, three for each colony.  My previous rules held that the tile draw would trigger game end when all seven colonies had at least two tiles.  But the Congress of Gamers playtest ran the maximum possible length, when the second China tile didn't come out until all 18 tiles on the other six colonies had been drawn.  That turned out to be too long.

So the latest adjustment I made was to add another game-end trigger - when any four colonies have three tiles each.  I tested this new condition for the first time last night, and as it happens, the game lasted the maximum possible number of tiles again - this time, three colonies had three tiles each, three colonies had two tiles each, and China had one tile before the 17th tile was drawn to trigger game end.  After the game, one of the playtesters suggested that I actually run the numbers to find out the expected game length and what the odds were that the game would run anywhere from the minimum (12) to maximum (17) number of tiles before triggering game end.

I've had a graduate-level course in combinatorics, so I thought this problem would be a piece of cake.  Well, not entirely.  I figured out what I thought were all the possible cases and the corresponding probabilites for the end-game conditions, but they only added up to 98%, which means I'm not accounting for something somewhere.  But in any event, here is the distribution I came up with, and I made a fascinating discovery:

No. of tiles Probability of game end
12 0.01%
13 0.14%
14 1.06%
15 11.70%
16 29.61%
17 55.73%
Total 98.25%

So, if my arithmetic is anywhere near correct, the game will probably (56%) go its maximum length at 17 tiles.  So I may continue to consider other options for triggering game end so that I get a less skewed distribution of game durations.

Based on this distribution, though, and the observed average of 3.7 minutes per player per turn, it appears that my game in its current instantiation will run 100 to 200 minutes, depending on the number of players.  Still a little longer than I like, but better than it was.

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