Backgammon Variants

Bearoff Wastage

Wastage refers to the difference between (1) the distance your checkers must move to come home and bear off and (2) the number of pips you expect have to roll to get your checkers off.

Adding wastage to your pip count gives your effective pip count (epc).

The term wastage traditionally applies in no-contact positions and refers to pips that go unused when you roll a number too high to be used to bear off a checker exactly.

I would now like to extend the meaning of wastage, so I will refer to traditional wastage as “bearoff wastage.”

Contact Wastage

The concept of wastage can be extended to contact positions. The basic idea is the same:

But the wastage in contact positions doesn’t come from bearing off; it comes from being blocked and being hit.

Block Wastage

Block wastage refers to pips you roll that you can’t use because your checkers are blocked. Here is an example.

If black is unlucky enough to roll 1-1, he won’t be able to move at all and his 4 rolled pips will be wasted. The immediate cost of this possibility is 4 pips in 36 rolls, or 0.11111 pips on average.

Here is another example.

There are many ways black can be blocked here. Rolling 6-6 loses 24 pips, 3-3 loses 9 pips, 2-2 loses 4 pips, 4-2 loses 2 pips, and 5-1 loses 1 pip. That’s a total of 43 pips out of 36 rolls, for an average of 1.1944 pips that black can expect to lose on his next roll.

In these two examples, we calculated black’s immediate block wastage. Black’s total block wastage is higher than that because he might be blocked again later in the game.

• Black’s total block wastage in Example 1 is 0.11137.
• Black’s total block wastage in Example 2 is 1.19990.

Hit Wastage

Hit wastage refers to pips you expect to lose from having your blots hit and sent to the bar. Hit wastage depends on: (1) how likely you are to be hit and (2) how far your blots are sent back when they are hit.

(It may seem a strange use of the term “wastage” to quantify the danger of being hit, but it is convenient to group hit wastage, block wastage, and bearoff wastage together. Pips lost to a hit must be made up by throwing additional rolls later so in some sense you are wasting pips by leaving blots where they can be hit.)

Here is an example of hit wastage:

On black’s next roll, he will leave a blot with any roll containing a 1 except 1-1. That’s 10/36 rolls. When black leaves a blot, white will hit it with 11 of his 36 rolls. When black is hit, his checker is sent back 23 pips. Let’s multiply that out:

Black expects to lose 1.95216 pips on average over the next exchange of rolls.

Black’s total hit wastage in this position is higher than that because he might be hit again later in the game, maybe several times. His total hit wastage in Example 3 is 3.08619 pips.

Defining Best Play

There is one further detail we need to nail down to be able to measure contact wastage in a consistent manner: How exactly will the game be played out?

Generally we assume both sides play to maximize their own equity. But sometimes there is a tie — sometimes two or more plays have the same equity. How do we break ties? Here is the policy I will use:

If two plays are tied in equity, choose the play that minimizes total epc.

We saw an example in Hyper 08 of where tie-breaking is needed:

It’s black’s turn and he has a choice of playing 3/1*/off (hitting) or 3/off (not hitting). Both choices yield the same equity (3 points for black). So to break the tie, we look at total epc. Black’s epc is the same either way. But white’s epc is higher when black hits, so we break the tie by choosing the nonhitting play for black.

Computing Wastage

A player’s total wastage is the sum of his bearoff wastage, his block wastage, and his hit wastage:

The wastage of a given position can be computed in two parts: Take the current wastage of the position and add the future wastage of the position.

The current wastage of a position comes from the next roll. For each of the 36 possible rolls, we observe the difference between the number of pips that were rolled and the number of pips that were played. If a blot was hit, charge the lost pips to the opponent’s wastage.

Future wastage is everything else, all the wastage after the next roll.

wastage of current position =
average wastage of next roll + average wastage of the 36 resulting positions

Wastage in Hypergammon

The following chart shows the average wastage you can expect to see in a game of hypergammon. “Player 1” is the player that wins the opening roll. “Player 2” is the player that loses the opening roll.

There are many tidbits of information to be gleaned from this chart.

1. The average bearoff wastage per player in a game of hypergammon game is 6.62318 pips. This is less than it is in backgammon. (Backgammon’s minimum bearoff wastage with 15 checkers is 7.06895, so the average wastage would be something higher than that.) That’s as you’d expect; every additional checker on the board is one more that can get caught up in a wasteful roll at the end of the game.

2. Block wastage in hypergammon is very low, less than half a pip per player during the whole game. Blocking is not a big factor in hypergammon. After all, your best block is only a single point and that’s not going to slow down your opponent very much.

3. Hit wastage makes up the vast majority of the wastage in hypergammon. Eighty-two percent of all wastage comes from blots being hit.

4. There is a curious number in this chart that needs explanation. In the column “First Roll,” you see the number 1.16667. This is a kludge to account for the fact that the first roll of a game cannot be doublets.

   The epc formula assumes an average of 8.16667 pips per roll, but that doesn’t hold for the first roll of the game, which averages just 7 pips. To make up the difference, we add 1.16667 pips to Player 1’s wastage.

5. The total wastage per player in hypergammon is 41.57749 pips. Your effective pip count (epc) is the sum of your regular pip count and your wastage. Since your initial pip count (pc) is 24 + 23 + 22 = 69, your effective pip count is:
   epc = pc + wastage = 69 + 41.57749 = 110.57749

   Effective pip count can be converted to number of rolls:

   epc ÷ (49/6) = 13.45010 rolls

   That means, from the start of a game of hypergammon, it takes on average 13.45010 rolls to bring your checkers home and bear them off.

(To be continued.)

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