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Math of backgammon

Posted By: Kye Hedlund
Date: Thursday, 9 December 2010, at 6:27 p.m.

In Response To: Math of backgammon (Sebastiaan Kuijs)

Match Equity Tables for Beginners

When first learning your match equities, the following simplified tables may be of use. I find rote memorization tedious and error prone. More useful to me is to thoroughly learn a simplified version that captures the essential ideas but omits many of the details. Then, once the simplified version is second nature to me, I refine my understanding little by little by filling in details. The Super Simplified MET is easily mastered. All entries are liberally rounded giving a simple and regular pattern. Learning it requires remembering only the four facts listed below the table.

Super Simplified MET

–2 –3 –4 –5 –6 –1C 70 75 80 85 90 –2 60 70 75 80 –3 60 65 70 –4 60 65

1) Crawford & lead by 1 = 70% 2) Non-Crawford & lead by 1 = 60% 3) +5% for each additional point leading 4) Exception at –2–4

Once this table is mastered, we can refine our understanding by making four adjustments. In the Step 2 table, all entries are accurate to within 1%. For most intermediate level players, this is probably accurate enough for over the board play. For comparison, the full Rockwell-Kazaross table is given below.

Super Simplified MET - Step 2

–2 –3 –4 –5 –6 –1C 68 75 80 85 90 –2 60 68 75 80 –3 58 65 70 –4 58 65 Adjust downward by 2% at

–1C–2

–2–4

–3–4

–4–5

All entries accurate to +- 1%

Rockwell–Kazaross MET

–2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –1C 68 75 81 84 89 91 93 94 96 97 –2 60 67 74 80 84 88 90 92 94 –3 57 65 71 76 80 84 87 89 –4 58 64 70 75 79 82 85 –5 57 63 68 73 77 80 –6 56 62 67 71 75 –7 55 61 66 70 –8 55 60 65 –9 55 60 –10 55

Understanding Match Equities

There are two fundamental factors that determine match equity: 1. Size of lead 2. How close you are to the end of the match

A lead of a single point gives an equity of about 60%. Each additional point adds another 5%. So, at –3–4 the leader has 60% match equity. 65% at –3–5. At –3–6, 70%, etc.

The Crawford game takes the cube out of your opponent’s hands which boosts the leader's equity by 10%. Thus, a one point lead at Crawford (–1C–2) has 70% equity rather than 60%, 2 points (–1C–3) 75% rather than 65%, etc.

At 2 away, you get an equity boost of 5%. Thus, a two point lead at –2–4 has an equity of 70% (compared to 65% at –3–5), –2–5 has 75% rather than 70%, etc. There are several reasons that it is advantageous to be only 2 away. The leader is: • Threatening to reach Crawford • Threatening to win the match with a gammon • The leader will rarely doubling making it difficult for the trailer to catch up by more than 1 or 2 points. • The leader's takepoint is lowered to 18% so it will be a little harder for the trailer to double him out.

(NOTE that the “2 away rule” does not apply does not apply to –2–3 which has the “standard” equity of 60%. I suspect that this is an artifact that results, in part, from my rounding. At farther than 2 away, a one point lead is really worth 58% or less, not the 60% as in the Super Simplified MET. Seen in this light, being 2 away is worth a 2% equity boost for a one point lead (–2–3 worth 60% not 58%), 3% for a two point lead (–2–4 worth 68% not 65%) and 5% at larger leads (–2–5 worth 75% not 70%).)

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