Very nice set of posts (although I'll need more time to go through Brad's Part B)!
In the discussion of "fairness" you mention the the number of tied games was surprisingly large. I also found the number of ties in my analysis of the Mancala game surprisingly large. I think we both draw that conclusion because we're both used to playing the game to win rather than exploring all possible games.
I downloaded the notebook and did a little exploration. The number of terminal game states (outs
in the PathCount
function) is quite large: 1988. Or more than 20% of the vertices in the canonical game graph of 9617 vertices. Only 567 of these terminal game states completely fill the game board, and the least filled game board has eight empty squares.
Have you explored other characteristics of the game, such as the distribution of the final scores, or the distribution of the number of flipped pieces, or the distribution of score reversals?
The number of pieces of each color for a given game state defines a point in the first quadrant of the Euclidean plane, so a game could be plotted as the scores along a path from the root vertex to some terminal vertex. The same technique could also be used to visualize the distribution of final game scores with Histogram3D
.
I'm glad to see that you would like to examine different game rules. That was very enlightening for the Mancala game.
The operation of the PathCount
function is a bit obscure. Perhaps a small worked example would help readers follow the logic.
I also found the plot of the frequencies of the path counts confusing. The association pcData
has an element 4 -> 126
, what does the 4
represent (there are 126 4
s in the output of PathCount
)?
I'm sure there are a great many papers published on this game and the larger version Othello. Do you know if there are any unanswered questions in these papers that might be attacked with multiway analysis?