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PR or Luck? (Time management)

Posted By: Chris Yep
Date: Saturday, 13 February 2016, at 5:46 a.m.

In Response To: PR or Luck? (Time management) (Rick Janowski)

Here are some examples illustrating Bob's point:

First, here's a MET for Bob's game (cubeless backgammon, no gammons/backgammons):

Cubeless Backgammon, No Gammons/Backgammons
Away/Away 1-away 2-away 3-away 4-away 5-away 6-away 7-away 8-away 9-away 10-away 11-away
1-away 50.00% 75.00% 87.50% 93.75% 96.88% 98.44% 99.22% 99.61% 99.80% 99.90% 99.95%
2-away 25.00% 50.00% 68.75% 81.25% 89.06% 93.75% 96.48% 98.05% 98.93% 99.41% 99.68%
3-away 12.50% 31.25% 50.00% 65.63% 77.34% 85.55% 91.02% 94.53% 96.73% 98.07% 98.88%
4-away 6.25% 18.75% 34.38% 50.00% 63.67% 74.61% 82.81% 88.67% 92.70% 95.39% 97.13%
5-away 3.13% 10.94% 22.66% 36.33% 50.00% 62.30% 72.56% 80.62% 86.66% 91.02% 94.08%
6-away 1.56% 6.25% 14.45% 25.39% 37.70% 50.00% 61.28% 70.95% 78.80% 84.91% 89.49%
7-away 0.78% 3.52% 8.98% 17.19% 27.44% 38.72% 50.00% 60.47% 69.64% 77.28% 83.38%
8-away 0.39% 1.95% 5.47% 11.33% 19.38% 29.05% 39.53% 50.00% 59.82% 68.55% 75.97%
9-away 0.20% 1.07% 3.27% 7.30% 13.34% 21.20% 30.36% 40.18% 50.00% 59.27% 67.62%
10-away 0.10% 0.59% 1.93% 4.61% 8.98% 15.09% 22.72% 31.45% 40.73% 50.00% 58.81%
11-away 0.05% 0.32% 1.12% 2.87% 5.92% 10.51% 16.62% 24.03% 32.38% 41.19% 50.00%

Consider four players:

A: Plays perfectly
B1: Makes a 0.2 EMG error at the beginning of game 1 (immediately before the "opening roll"), reducing his game 1 winning chance from 50% to 40%; otherwise plays perfectly
B2: Makes a 0.2 EMG error at the beginning of game 2; otherwise plays perfectly
B3: Makes a 0.2 EMG error at the beginning of game 3; otherwise plays perfectly

In an 11-point match, what are A's winning chances against B1, B2, and B3 respectively? It turns out that A has a 51.762% winning chance against each of the three imperfect players. Despite the fact that B3 always makes a blunder later in the match (in game 3), he still has a 48.238% chance of beating A in an 11-point match, the exact same chance that B1 and B2 have of beating A in an 11-point match.

In a 5-point match, each of the three imperfect players has a 47.266% chance of beating A.

Calculations (refer to the color-coded MET below):

Cubeless Backgammon, No Gammons/Backgammons
Away/Away 1-away 2-away 3-away 4-away 5-away 6-away 7-away 8-away 9-away 10-away 11-away
1-away 50.00% 75.00% 87.50% 93.75% 96.88% 98.44% 99.22% 99.61% 99.80% 99.90% 99.95%
2-away 25.00% 50.00% 68.75% 81.25% 89.06% 93.75% 96.48% 98.05% 98.93% 99.41% 99.68%
3-away 12.50% 31.25% 50.00% 65.63% 77.34% 85.55% 91.02% 94.53% 96.73% 98.07% 98.88%
4-away 6.25% 18.75% 34.38% 50.00% 63.67% 74.61% 82.81% 88.67% 92.70% 95.39% 97.13%
5-away 3.13% 10.94% 22.66% 36.33% 50.00% 62.30% 72.56% 80.62% 86.66% 91.02% 94.08%
6-away 1.56% 6.25% 14.45% 25.39% 37.70% 50.00% 61.28% 70.95% 78.80% 84.91% 89.49%
7-away 0.78% 3.52% 8.98% 17.19% 27.44% 38.72% 50.00% 60.47% 69.64% 77.28% 83.38%
8-away 0.39% 1.95% 5.47% 11.33% 19.38% 29.05% 39.53% 50.00% 59.82% 68.55% 75.97%
9-away 0.20% 1.07% 3.27% 7.30% 13.34% 21.20% 30.36% 40.18% 50.00% 59.27% 67.62%
10-away 0.10% 0.59% 1.93% 4.61% 8.98% 15.09% 22.72% 31.45% 40.73% 50.00% 58.81%
11-away 0.05% 0.32% 1.12% 2.87% 5.92% 10.51% 16.62% 24.03% 32.38% 41.19% 50.00%

Match Winning Probabilities:

A vs. B1, 11-point match:
Pr(B1 win) = (0.4)(.58810) + (0.6)(.41190) = 48.238%

A vs. B2, 11-point match:
Pr(B2 win) = (0.5)(0.4 * 0.5 + 0.6 * 0.32380) + (0.5)(0.4 * 0.67620 + 0.6 * 0.5) = 48.238%

A vs. B3, 11-point match:
Pr(B3 win) = (0.25)(0.4 * 0.40726 + 0.6 * 0.24034) + (0.5)(0.4 * 0.59274 + 0.6 * 0.40726) + (0.25)(0.4 * 0.75966 + 0.6 * 0.59274) = 48.238%

A vs. B1, 5-point match:
Pr(B1 win) = (0.4)(0.63672) + (0.6)(0.36328) = 47.266%

A vs. B2, 5-point match:
Pr(B2 win) = (0.5)(0.4 * 0.5 + 0.6 * 0.22656) + (0.5)(0.4 * 0.77344 + 0.6 * 0.5) = 47.266%

A vs. B3, 5-point match:
Pr(B3 win) = (0.25)(0.4 * 0.34375 + 0.6 * 0.10938) + (0.5)(0.4 * 0.65625 + 0.6 * 0.34375) + (0.25)(0.4 * 0.89063 + 0.6 * 0.65625) = 47.266%

In each of the 11-point match examples, an average of 17.62% MWC is at stake (defined as MWC(win game) - MWC(lose game)) in the game that the imperfect player blunders.

A vs. B1: (58.81% - 41.19%) = 17.62% MWC is at stake in game 1.
A vs. B2: 50% of the time (50% - 32.38%) = 17.62% MWC is at stake in game 2; 50% of the time 67.62% - 50% = 17.62% MWC is at stake in game 2; on average 17.62% MWC is at stake in game 2.
A vs. B3: 25% of the time (40.73% - 24.03%) = 16.70% MWC is at stake in game 3; 50% of the time (59.27% - 40.73%) = 18.54% MWC is at stake in game 3; 25% of the time (75.97% - 59.27%) = 16.70% MWC is at stake in game 3; on average 17.62% MWC is at stake in game 3.

Similarly, in each of the 5-point match examples, an average of 27.34% MWC is at stake in the game that the imperfect player blunders.

A vs. B1: (63.67% - 36.33%) = 27.34% MWC is at stake in game 1.
A vs. B2: (50%)(50% - 22.66%) + (50%)(77.34% - 50%) = 27.34% MWC (amount at stake in game 2)
A vs. B3: (25%)(34.38% - 10.94%) + (50%)(65.63% - 34.38%) + (25%)(89.06% - 65.63%) = 27.34% MWC (average amount at stake in game 3)

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