Match Equities |

*Backgammon with the Giants: Neil Kazaross*,

by Antonio Ortega and Danny Kleinman.

We include below the match equity chart that we will use in this book. It agrees substantially with Kit Woolsey’s empirically-derived chart, but contains winning chances to three decimal places, a refinement we think necessary. The third decimal place has been obtained by adjusting Woolsey’s two-decimal-place probabilities in the direction of the theoretical probabilities that Danny calculated many years ago, which differ markedly little from Woolsey’s.

Where the two probabilities differ by some number of thousandths, *n*, we have adjusted Woolsey’s figures by the square root of *n* rounded to the nearest integer and divided by 1000. For example, at (needs 4, needs 7), Woolsey’s chart shows .70 winning chances while Danny’s chart shows .711. The difference is 11 thousandths. Adjusting Woolsey’s figure by the square root of 11 (which is approximately 3), divided by 1000, we get .703, the probability that appears in the new chart.

The new chart is entirely consistent with Woolsey’s, no entry deviating by more than .005. For scores very early in the match and in the Crawford game, which are not covered in Danny’s earlier chart, we have made new estimates that are compatible with other entries.

PC | C | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |

PC | .500 | .510 | .700 | .710 | .830 | .835 | .902 | .905 | .944 | .946 | .968 | .969 | .982 | .983 | ||

C | .500 | .700 | .750 | .831 | .852 | .902 | .913 | .942 | .951 | .967 | .972 | .981 | .984 | .989 | .991 | |

2 | .490 | .300 | .500 | .598 | .678 | .748 | .808 | .848 | .880 | .908 | .927 | .944 | .955 | .965 | .972 | .978 |

3 | .300 | .250 | .402 | .500 | .589 | .658 | .715 | .764 | .805 | .844 | .871 | .897 | .915 | .935 | .946 | .959 |

4 | .290 | .169 | .322 | .411 | .500 | .580 | .644 | .703 | .754 | .794 | .829 | .857 | .881 | .900 | .917 | .930 |

5 | .170 | .148 | .252 | .342 | .420 | .500 | .573 | .633 | .684 | .733 | .774 | .807 | .835 | .865 | .885 | .900 |

6 | .165 | .098 | .192 | .285 | .356 | .427 | .500 | .562 | .623 | .673 | .715 | .756 | .788 | .824 | .848 | .869 |

7 | .098 | .087 | .152 | .236 | .297 | .367 | .438 | .500 | .562 | .613 | .656 | .705 | .741 | .777 | .806 | .836 |

8 | .095 | .058 | .120 | .195 | .246 | .316 | .377 | .438 | .500 | .553 | .604 | .654 | .693 | .730 | .767 | .801 |

9 | .056 | .049 | .092 | .156 | .206 | .267 | .327 | .387 | .447 | .500 | .553 | .603 | .645 | .684 | .723 | .757 |

10 | .054 | .033 | .073 | .129 | .171 | .226 | .285 | .344 | .396 | .447 | .500 | .552 | .598 | .638 | .679 | .712 |

11 | .032 | .028 | .056 | .103 | .143 | .193 | .244 | .295 | .346 | .397 | .448 | .500 | .552 | .591 | .626 | .666 |

12 | .031 | .019 | .045 | .085 | .119 | .165 | .212 | .259 | .307 | .355 | .402 | .448 | .500 | .544 | .585 | .624 |

13 | .018 | .016 | .035 | .065 | .100 | .135 | .176 | .223 | .270 | .316 | .362 | .409 | .456 | .500 | .544 | .582 |

14 | .017 | .011 | .028 | .054 | .083 | .115 | .152 | .194 | .233 | .277 | .321 | .374 | .415 | .456 | .500 | .540 |

15 | .009 | .022 | .041 | .070 | .100 | .131 | .164 | .199 | .243 | .288 | .334 | .376 | .418 | .460 | .500 |