Jake Jacobs and Walter Trice, 1996
Can a Fish Taste Twice as Good?, © 1996 Jake Jacobs and Walter Trice

Taste Twice as Good?
, by Jake Jacobs and Walter Trice.
Visit Jacobs’s web site at The Doubling Cube.

If there is a heart to this book, it is Tables 13–23. Table 13 is our benchmark table — use it for any even match up to 25 points in length. It was created by Walter’s program, MEG3, which stands for “match equity generator.” MEG takes a number of parameters:

• The CPW is the “cubeless probability of winning.” When two players are evenly matched, this is 0.5 for both.

• The gammon rate has been the subject of much controversy ever since Roy Friedman rejected the gammon rated used by Danny Kleinman in developing his chart. Danny estimated that 20% of each player’s wins were gammons. Roy claimed that extensive rollouts by him had consistently shown gammon rates much higher than 20%, possibly as high as 36%.

Danny’s 20% seems too low; Roy’s 36% seems much too high. Two sources have provided objective data in recent years. One is Hal Heinrich’s database of matches. Simply count the gammons, divide by the number of games, and voila! Hal’s numbers suggest a 26% gammon rate. It is also consistent with results from the second objective source, computer rollouts.

Some feel this is still slightly high. Many of Hal’s matches were played in the 1970’s, when playing styles were different. Modern computer rollouts may be biased by the biases of the programs. Kit Woolsey has told me that he personally wins 26% gammons and suggested that most players at all skill levels do not play for enough gammons. I suspect he is correct, but Kit is stronger than most of his opponents and may win more gammons for that reason. I have settled on 24% for the standard table.

• The backgammon rate is set at 1%. I doubt that the backgammon rate between players will ever vary by as much as 1% — backgammons are too rare — and that variance is not enough to affect the table, so the backgammon rate is the same throughout.

• The Type-1 overshoot factor adjusts for the player’s inability to give perfectly timed doubles. This inability is, to some extent, inherent in the game, but it increases as the player’s own lack of ability increases.

Using a one-checker model, Walter is able to demonstrate that the inherent Type-1 overshoot ranges from 5%, when the cube owner’s CPW is near 50%, to 8%, when his CPS is down near 20%. I have set the Type-1 overshoot at 0.06.

• The Type-2 overshoot deals specifically with the player’s match doubles. Errors are due, not to misevaluation of the position, but to misevaluation of the demands of the match score. Typically, Type-2 overshoot acts as an intensifier for Type-1’s distortions.

On its own, it has little effect on cube handling until its setting is greater than 1.0. A setting of 1.0 implies that the players is always doubling as though playing for money; beyond 1.0, he is handling the cube less accurately than for money. Type-2 overshoot is set at 0.10.

Walter believes both Type-1 and Type-2 overshoot might be set even higher, but I have cleaved to the conservative path.

• The gammon reduction factor, which reduces the assumed gammon rate for each higher notch on the cube, is set at 0.5. No study has determined precisely what the gammon reduction should be. Observation suggest that many higher cubes arise during the bearoff, or after a late shot is hit. No gammons are involved, so the 24% gammon rate is too high for big cubes.

With a gammon reduction of 0.5, the gammon rate drops from 24% to 12%, then to 6%, etc., for each new cube level. This should be close to reality, and small changes have negligible affect.

• Equity at 2-away, 2-away. I have set this at 50%. Most match players are familiar with the theory that the cube should always be turned before one’s opponent has a pass when the score is 2-away, 2-away.

Parameters used to create Table 13
 Player A Player B Cubeless probabilityof winning (CPW) 0.5 0.5 Gammon rate 0.24 0.24 Backgammon rate 0.01 0.01 Type 1 overshoot 0.06 0.06 Type 2 overshoot 0.10 0.10 Gammon reduction rate 0.5 0.5 Maximum score 25 25 Equity at 2-away, 2-away 0.5 0.5

Table 13:
Match Equity between Equal Players
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 0.5 0.69 0.748 0.822 0.844 0.894 0.909 0.938 0.946 0.963 0.968 0.978 0.981 0.987 0.989 0.992 0.994 0.996 0.996 0.997 0.998 0.998 0.999 0.999 0.999 2 0.31 0.5 0.603 0.686 0.753 0.81 0.851 0.886 0.91 0.931 0.946 0.959 0.967 0.975 0.98 0.985 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.998 0.999 3 0.252 0.397 0.5 0.579 0.654 0.719 0.769 0.814 0.848 0.879 0.902 0.922 0.937 0.95 0.96 0.969 0.975 0.981 0.984 0.988 0.99 0.993 0.994 0.995 0.996 4 0.178 0.314 0.421 0.5 0.579 0.647 0.704 0.753 0.795 0.831 0.861 0.886 0.907 0.925 0.939 0.951 0.96 0.968 0.974 0.979 0.983 0.987 0.989 0.992 0.993 5 0.156 0.247 0.346 0.421 0.5 0.568 0.629 0.683 0.73 0.773 0.808 0.84 0.866 0.889 0.908 0.924 0.938 0.949 0.958 0.966 0.972 0.978 0.982 0.985 0.988 6 0.106 0.19 0.281 0.353 0.432 0.5 0.563 0.62 0.672 0.718 0.759 0.795 0.826 0.853 0.876 0.896 0.913 0.928 0.94 0.95 0.959 0.966 0.972 0.977 0.981 7 0.091 0.149 0.231 0.296 0.371 0.437 0.5 0.557 0.611 0.66 0.704 0.744 0.779 0.811 0.838 0.862 0.883 0.901 0.917 0.93 0.941 0.951 0.959 0.966 0.972 8 0.062 0.114 0.186 0.247 0.317 0.38 0.443 0.5 0.555 0.606 0.653 0.695 0.734 0.768 0.799 0.827 0.851 0.873 0.891 0.907 0.921 0.934 0.944 0.953 0.961 9 0.054 0.09 0.152 0.205 0.27 0.328 0.389 0.445 0.5 0.551 0.6 0.644 0.685 0.723 0.757 0.788 0.816 0.84 0.862 0.881 0.898 0.913 0.926 0.937 0.946 10 0.037 0.069 0.121 0.169 0.227 0.282 0.34 0.394 0.449 0.5 0.549 0.595 0.638 0.678 0.715 0.748 0.779 0.806 0.831 0.853 0.872 0.89 0.905 0.918 0.93 11 0.032 0.054 0.098 0.139 0.192 0.241 0.296 0.347 0.4 0.451 0.5 0.546 0.591 0.632 0.671 0.706 0.739 0.769 0.797 0.821 0.844 0.864 0.881 0.897 0.911 12 0.022 0.041 0.078 0.114 0.16 0.205 0.256 0.305 0.356 0.405 0.454 0.5 0.545 0.587 0.627 0.665 0.7 0.732 0.761 0.789 0.813 0.835 0.856 0.874 0.89 13 0.019 0.033 0.063 0.093 0.134 0.174 0.221 0.266 0.315 0.362 0.409 0.455 0.5 0.543 0.584 0.622 0.659 0.693 0.725 0.754 0.781 0.805 0.828 0.848 0.866 14 0.013 0.025 0.05 0.075 0.111 0.147 0.189 0.232 0.277 0.322 0.368 0.413 0.457 0.5 0.541 0.581 0.618 0.654 0.687 0.718 0.747 0.774 0.798 0.82 0.841 15 0.011 0.02 0.04 0.061 0.092 0.124 0.162 0.201 0.243 0.285 0.329 0.373 0.416 0.459 0.5 0.54 0.578 0.614 0.649 0.682 0.712 0.741 0.767 0.791 0.814 16 0.008 0.015 0.031 0.049 0.076 0.104 0.138 0.173 0.212 0.252 0.294 0.335 0.378 0.419 0.46 0.5 0.539 0.576 0.611 0.645 0.677 0.707 0.735 0.761 0.785 17 0.006 0.012 0.025 0.04 0.062 0.087 0.117 0.149 0.184 0.221 0.261 0.3 0.341 0.382 0.422 0.461 0.5 0.537 0.573 0.608 0.641 0.672 0.702 0.729 0.755 18 0.004 0.009 0.019 0.032 0.051 0.072 0.099 0.127 0.16 0.194 0.231 0.268 0.307 0.346 0.386 0.424 0.463 0.5 0.536 0.571 0.605 0.637 0.668 0.697 0.724 19 0.004 0.007 0.016 0.026 0.042 0.06 0.083 0.109 0.138 0.169 0.203 0.239 0.275 0.313 0.351 0.389 0.427 0.464 0.5 0.535 0.57 0.602 0.634 0.664 0.692 20 0.003 0.005 0.012 0.021 0.034 0.05 0.07 0.093 0.119 0.147 0.179 0.211 0.246 0.282 0.318 0.355 0.392 0.429 0.465 0.5 0.534 0.568 0.6 0.631 0.66 21 0.002 0.004 0.01 0.017 0.028 0.041 0.059 0.079 0.102 0.128 0.156 0.187 0.219 0.253 0.288 0.323 0.359 0.395 0.43 0.466 0.5 0.534 0.566 0.598 0.628 22 0.002 0.003 0.007 0.013 0.022 0.034 0.049 0.066 0.087 0.11 0.136 0.165 0.195 0.226 0.259 0.293 0.328 0.363 0.398 0.432 0.466 0.5 0.533 0.565 0.596 23 0.001 0.002 0.006 0.011 0.018 0.028 0.041 0.056 0.074 0.095 0.119 0.144 0.172 0.202 0.233 0.265 0.298 0.332 0.366 0.4 0.434 0.467 0.5 0.532 0.563 24 0.001 0.002 0.005 0.008 0.015 0.023 0.034 0.047 0.063 0.082 0.103 0.126 0.152 0.18 0.209 0.239 0.271 0.303 0.336 0.369 0.402 0.435 0.468 0.5 0.531 25 0.001 0.001 0.004 0.007 0.012 0.019 0.028 0.039 0.054 0.07 0.089 0.11 0.134 0.159 0.186 0.215 0.245 0.276 0.308 0.34 0.372 0.404 0.437 0.469 0.5