The next step after getting the Crawford-game match equities is to build out the full match equity tables for pre-Crawford games.

I followed the algorithm defined here, by Tom Keith, but there were a few assumptions I didn't fully understand. More on that later. Here is my calculated pre-Crawford match equity table, out to 12-away:

And here are the corresponding post-Crawford match equities, recalculated using 2-ply Player 3.3 to get state probabilities in the first two rolls.

I calculated the average probability of win, gammon win, and backgammon win, and corrected all player-calculated probabilities so that the average probability of win was 50% and the average probability of gammon win/loss and backgammon win/loss were symmetric.

The probability of the game ending in gammon using these adjusted state game probabilities is 27.60%. The probability of the game ending in backgammon is 1.1%.

The approach:

This converged to exactly the values described by the Keith algorithm above. So the Keith algorithm is equivalent to a live cube model where probabilities diffuse continuously rather than jump. This makes me wonder whether a more accurate match equity table could mix between this limit and something like the dead cube limit, as with the money cube algorithm described by Janowski.

I followed the algorithm defined here, by Tom Keith, but there were a few assumptions I didn't fully understand. More on that later. Here is my calculated pre-Crawford match equity table, out to 12-away:

1-away | 2-away | 3-away | 4-away | 5-away | 6-away | 7-away | 8-away | 9-away | 10-away | 11-away | 12-away | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1-away | 0 | 0.362 | 0.510 | 0.635 | 0.690 | 0.780 | 0.820 | 0.869 | 0.892 | 0.922 | 0.936 | 0.953 |

2-away | -0.362 | 0 | 0.189 | 0.322 | 0.472 | 0.585 | 0.667 | 0.733 | 0.788 | 0.834 | 0.867 | 0.896 |

3-away | -0.510 | -0.189 | 0 | 0.139 | 0.289 | 0.415 | 0.513 | 0.595 | 0.667 | 0.729 | 0.777 | 0.819 |

4-away | -0.635 | -0.322 | -0.139 | 0 | 0.149 | 0.279 | 0.385 | 0.474 | 0.558 | 0.630 | 0.690 | 0.741 |

5-away | -0.690 | -0.472 | -0.289 | -0.149 | 0 | 0.131 | 0.245 | 0.343 | 0.436 | 0.518 | 0.588 | 0.649 |

6-away | -0.780 | -0.585 | -0.415 | -0.279 | -0.131 | 0 | 0.118 | 0.221 | 0.320 | 0.409 | 0.488 | 0.557 |

7-away | -0.820 | -0.667 | -0.513 | -0.385 | -0.245 | -0.118 | 0 | 0.105 | 0.208 | 0.300 | 0.385 | 0.460 |

8-away | -0.869 | -0.733 | -0.595 | -0.474 | -0.343 | -0.221 | -0.105 | 0 | 0.103 | 0.198 | 0.287 | 0.367 |

9-away | -0.892 | -0.788 | -0.667 | -0.558 | -0.436 | -0.320 | -0.208 | -0.103 | 0 | 0.096 | 0.188 | 0.271 |

10-away | -0.922 | -0.834 | -0.729 | -0.630 | -0.518 | -0.409 | -0.300 | -0.198 | -0.096 | 0 | 0.092 | 0.178 |

11-away | -0.936 | -0.867 | -0.777 | -0.690 | -0.588 | -0.488 | -0.385 | -0.287 | -0.188 | -0.092 | 0 | 0.087 |

12-away | -0.953 | -0.896 | -0.819 | -0.741 | -0.649 | -0.557 | -0.460 | -0.367 | -0.271 | -0.178 | -0.087 | 0 |

And here are the corresponding post-Crawford match equities, recalculated using 2-ply Player 3.3 to get state probabilities in the first two rolls.

1-away | 2-away | 3-away | 4-away | 5-away | 6-away | 7-away | 8-away | 9-away | 10-away | 11-away | 12-away | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1-away | 0 | 0.028 | 0.362 | 0.388 | 0.626 | 0.645 | 0.774 | 0.788 | 0.865 | 0.874 | 0.919 | 0.925 |

I calculated the average probability of win, gammon win, and backgammon win, and corrected all player-calculated probabilities so that the average probability of win was 50% and the average probability of gammon win/loss and backgammon win/loss were symmetric.

The probability of the game ending in gammon using these adjusted state game probabilities is 27.60%. The probability of the game ending in backgammon is 1.1%.

The approach:

- Define M(n,m) as the match equity for player n-away and opponent m-away, for symmetric players before the first dice are thrown in a game (so both players have 50% chance of a win and the cube is at 1). By convention I'll assume n<m but M(m,n)=-M(n,m) so this isn't an issue.
- Define Mp(n,m,c,X,Pw) as the match equity for player n-away, opponent m-away, cube value c, owned by player X (either P for player, O for opponent, or C for centered when c=1), and probability of any player win Pw. We'll assume this is piecewise linear in Pw in the range [take point,cash point] for centered cube, [take point,1] for opponent-owned cube, and [0,cash point] for player-owned cube.
- The match equity table value M(n,m) = Mp(n,m,1,C,0.5), so if we can get Mp functions we're done.
- We start with the cube at the maximum value, so that c is the smallest value that's greater than m (remember we assume m>n by convention). In this state the player owns the cube, since only the opponent can double to this level. No more doubling can happen, and any win or loss is a match win or loss, so Mp(n,m,cmax,P)=2 Pw - 1.
- Then we step back to the cube 2x smaller than the maximum. For the case where the player owns the cube we calculate the cash point, where the opponent's equity in the maximum cube state equals their equity on a pass (which uses the match equity table for the corresponding away values which we bootstrap). We assume that the player does not double until the cash point, and so M(n,m,c,P) is linear from Pw=0 to Pw=cash point. We know the cash point value from the opponent's take/pass decision, and we know the match equity for a player loss at this cube value (which also uses the appropriate values from the match equity table for M(n,smaller m values)).
- For the case when the opponent owns the cube, we assume the opponent doubles at the player's take point, and that the opponent's equity is piecewise linear in Pw from the take point to Pw=1 (where we know the opponent's winning value from the match equity table lookup).
- We continue to step back to cube=1, where we assume that match equity is linear between the player's take point and cash point. This gives us Mp(n,m,1,C,Pw), which we use to calculate M(n,m) = Mp(n,m,1,C,0.5).

This follows Tom Keith's algorithm and returns the match equity table displayed above, using the probability of gammon given above.

The main assumption here is that the player and opponent double only when they are at the cash point or take point. This seems a bit unusual - really you'd double earlier than the cash point, and the opponent would double you earlier than your take point.

I tried an alternative approach, similar to the "live cube" model in Janowski: the probability of win diffuses like a Brownian motion with constant volatility sigma. Then I can treat the doubling problem as a derivatives pricing problem, using numerical backward induction to get the fair value. Given a cube value I know the match equity at Pw=0 and Pw=1, and also that the match equity (given a cube value) satisfies the partial differential equation sigma^2/2 d^2(match equity)/dPw^2 + d(match equity)/dt = 0. You solve this PDE separately for different "layers" corresponding to different cube values and cube ownership, and at each time step mix the values across layers based on doubling decisions.

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