So 4/1 3/off(3) wins more often, wins fewer gammons, and -- big surprise? -- wins about 60% more backgammons, even though it gets hit 2/36 more times?
(1) 4/1 3/off(3) wins more often because the 12th checker off is huge. If you're hit, what's the worst that can happen? Suppose unrealistically that you never roll a quick joker (and win), Opponent has no trouble bringing his snake home (and you win), and he always makes a nice closed board bearoff:
- If you played 4/1(2) 3/off(2), you won right away 69.44% of the time. You got hit 30.55% of the time -- but of these, since you have only 11 checkers off, you now rate to win 75% of the time. 75% of 30.55% is 22.9%. 69.44% + 22.9% is 92.3%.
- If you played 4/1 3/off(3), you won right away 63.88% of the time. You got hit 30.55% plus 5.55% = 36.11% of the time -- but of these, with 12 checkers off you now rate to win 83%. 83% of 36.11% is 30%. 63.88% + 30% = 93.88%
So taking off 3 checkers wins about 1.5% more often, and of course you do have other ways to win (after either play) by rolling quick jokers or by jumping the snake.
(2) 4/1 3/off(3) wins fewer gammons for the obvious reason that almost the only way you can fail to win a gammon is by getting hit. 4/1 3/off(3) gets hit more often.
(3) 4/1 3/off(3) wins a lot more backgammons because:
Opponent will never keep the 23 point hoping you roll your one bad double. He will either hit or run both checkers. If he runs both checkers he sometimes saves the backgammon immediately. Otherwise:
- if you've played 4/1 3/off(3) you win a backgammon 5/36 of the time immediately, since 2-2 bears off, plus whenever Opponent's next roll doesn't save the backgammon.
- if you've played 4/1(2) 3/off(2) you win a backgammon only 4/36 of the time immediately, plus whenever Opponent's next roll doesn't save the backgammon. Of his rolls that neither immediately save the gammon or hit, his checkers will be slightly worse placed to save the backgammon.
| Opp's Roll | Your backgammon wins after 4/1 3/off(3) | and after 4/1(2) 3/off(3) | |
| 66 | 0 | 0 | |
| 65 | 0 | 0 | |
| 55 | 0 | 0 | |
| 44 | 0 | 0 | |
| 33 | 0 | 0 | |
| 21 | ~0 | ~0 | |
| 11 | ~0 | ~0 | |
| 61 | 5/36 + 1/36*13/36 + 30/36*2/36 | ~0 | |
| 51 | 5/36 + 1/36*13/36 + 30/36*2/36 | ~0 | |
| 31 | 5/36 + 1/36*13/36 + 30/36*13/36 | ~0 | |
| 41 | 5/36 + 1/36*13/36 + 30/36*5/36 | ~0 | |
| 64 | 5/36 | 4/36 | |
| 63 | 5/36 | 4/36 | |
| 54 | 5/36 | 4/36 | |
| 53 | 5/36 | 4/36 | |
| 43 | 5/36 | 4/36 | |
| 62 | ~0 | 4/36 | |
| 52 | ~0 | 4/36 | |
| 22 | ~0 | 4/36 | |
| 42 | ~0 | 4/36 + 1/36*13/36 + 31/36*5/36 | |
| 32 | ~0 | 4/36 + 1/36*13/36 + 31/36*5/36 |
Note that 1-1 picks and passes. And I've left out some extra backgammons from pick'n'passing with 2-1's after best play followed by 61 51 41 or 31, or after the other play followed by 42 or 32.
The score (after 0 games) is: White 0, Blue 0 (match to 7 points)
Move number 5: Blue to play 33
| White | 144 |
|---|---|
| |
| Blue | 18 |
Position ID: tK1BBRi5AQAAAA Match ID: QYntAAAAAAAE | |
• Blue moves 4/1 3/off(3)
| # | Ply | Move | Equity | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| • | 1 | R | 4/1 3/off(3) | +1.7960 | ||||||||||||||||||
| ||||||||||||||||||||||
| Full cubeful rollout with var.redn. | ||||||||||||||||||||||
| 1296 games, Mersenne Twister dice gen. with seed 929916246 and quasi-random dice | ||||||||||||||||||||||
| Play: world class 2-ply cubeful prune [world class] | ||||||||||||||||||||||
| keep the first 0 0-ply moves and up to 8 more moves within equity 0.16 | ||||||||||||||||||||||
| Skip pruning for 1-ply moves. | ||||||||||||||||||||||
| Cube: 2-ply cubeful prune [world class] | ||||||||||||||||||||||
| 2 | R | 4/1(2) 3/off(2) | +1.7335 ( -0.0625) | |||||||||||||||||||
| ||||||||||||||||||||||
| Full cubeful rollout with var.redn. | ||||||||||||||||||||||
| 1296 games, Mersenne Twister dice gen. with seed 929916246 and quasi-random dice | ||||||||||||||||||||||
| Play: world class 2-ply cubeful prune [world class] | ||||||||||||||||||||||
| keep the first 0 0-ply moves and up to 8 more moves within equity 0.16 | ||||||||||||||||||||||
| Skip pruning for 1-ply moves. | ||||||||||||||||||||||
| Cube: 2-ply cubeful prune [world class] | ||||||||||||||||||||||