Forum Archive : Terminology

"Gammon rate", "gammon price"

From:   David Montgomery
Address:   monty@cs.umd.edu
Date:   13 June 1995
Subject:   Re: Gammon rate and gammon price
Forum:   rec.games.backgammon
Google:   3rkk7o$lr4@twix.cs.umd.edu

USRobots writes:
> In various postings recently I've seen the terms
> "Gammon Rate" and "Gammon Price".  Can somebody explain them?

Gammon rate generally refers to the percentage of one's wins which
are gammons (including backgammons).  Typically it refers to cubeless
results, though this need not be so.  If no other context is provided,
it generally refers to the gammon rate of the opening position between
two equal players.

For example, a gammon rate in the opening position of 22% would mean
that 22% of each player's wins would be gammons, so that the
cubeless results would be (ignoring backgammons):

Player 1 wins plain game: 39%
Player 1 wins gammon:     11%
Player 2 wins plain game: 39%
player 2 wins gammon:     11%

Another example, not the opening position:

Player 1 wins plain game: 60%
Player 1 wins gammon:     15%
Player 2 wins plain game: 22%
player 2 wins gammon:      3%

Here player 1 has a 20% gammon rate (15/(60+15)) and player 2
has a 12% gammon rate (3/(22+3)).

The consensus is that the opening position has a gammon rate in the
low to mid 20's.   I think Kit Woolsey's analysis of games from
Hal Heinrich's match database gave a gammon rate of 21 or 22%.
Rollouts with Jellyfish have reportedly given a gammon rate of
around 26%.  Roy Friedman has argued for a gammon rate in the
mid 30's.  Expert Backgammon rollouts have given gammon rates in
the mid 20's (PC version) and low to mid 30's (Mac version).

The opening gammon rate is important for constructing match
equity tables.  Consider the score -1:-2 Crawford.  How often
will the trailer win?  Well, the trailer will win whenever
he or she wins a gammon, plus half the time when he or she
wins a plain game.  So, with a gammon rate of 20%, the trailer
will win 10+1/2*40=30% of the time.  With a gammon rate of 36%,
the trailer will win 18+1/2*32=34% of the time.

Note that the gammon rate depends on the match score, the relative
strengths of the players, and players' styles.  Thus it is somewhat
inaccurate to speak of the gammon rate of a particular position
without addressing these features, but this is what is commonly

The gammon rate can also refer simply to the probability of
winning a gammon, unadjusted for the probability of winning, if
it is clear that this is what is meant by the context.


The gammon price refers to value a gammon has beyond that of
a plain win, relative to the difference between winning and
losing a plain game.  That is:

                 (Equity from gammon win)-(Equity from plain game win)
Gammon Price = --------------------------------------------------------
                   (Equity from plain win) - (Equity from plain loss)

In money play, we have:

                  (+2*cube)  -  (+1*cube)
Gammon Price =  ---------------------------- = 0.5
                   (+1*cube) - (-1*cube)

This is what is meant when people say that gammons are half as
valuable as winning.  An important point for match play is that
a gammon price greater than 0.5 indicates that gammons are more
valuable than money play, while a gammon price of less than 0.5
indicates that gammons are less valuable than money play.

The gammon price can simplify the calculations for taking a double.
The procedure is this:

1) Estimate the probability of winning and losing a gammon.  Note
   that this is *not* the gammon rate, but the unadjusted probabilities.
2) Adjust your take point by adding your opponent's gammon probability
   multiplied by the opponent's gammon price, and subtracting your
   gammon probability multiplied by your gammon price.

If the gammon prices are the same, as they are in money play, then
step 2 is simpler:

2) Adjust your take point by adding the difference in the gammon
   probabilities multiplied by the gammon price.

For example, let's say you estimate the gammon probabilities as:

Opponent wins gammon:     15%
     You win  gammon:      3%

If your gammonless take point is 22% and the gammon price is 0.50,
then you need to win 22 + 0.50*(15-3)=28% to take here.

Knowledge of gammon prices can also simplify the decision to
play on for a gammon or cash.  If your gammons, multiplied by your
gammon price, minus your opponent's gammons, multiplied by their
gammon price, exceed your losing chances, then you should definitely
play on for the gammon.  In money play, this simplifies to checking
whether your excess gammons (your gammons minus your opponent's) are
at least twice your losing chances.

Calculation of gammon prices in match play uses the match winning
chances (mwc) at different scores.  For example, consider the score

With the cube on 1, the relevant mwc are (using Woolsey's table):

Leader wins gammon:     100%
Leader wins plain game:  85%
Trailer wins plain game: 60%
Trailer wins gammon:     50%

The leader's gammon price is (100-85)/(85-60) = 0.60.
The trailer's gammon price is (50-40)/(40-15) = 0.40.

Now consider with the cube on 2:

Leader wins gammon:     100%
Leader wins plain game: 100%
Trailer wins plain game: 50%
Trailer wins gammon:      0%

The leader's gammon price is (100-100)/(100-50) = 0.00.
The trailer's gammon price is (100-50)/(50-0)   = 1.00.

As can be seen here, the gammon prices in match play can vary
a great deal based on the score and the cube position.  Doubling
may raise or lower both players' gammon prices, and this can play
a big role in the correct doubling decisions.  For example, at
-2:-4, the leader has a big incentive not to double in a position
that is at all gammonish, since it lowers the leader's gammon price
from 0.60 to 0.00 while raising the trailer's gammon price from
0.40 to 1.00.  Conversely, the trailer should double very aggressively
if there is much chance for a gammon -- in fact, at this score the
opening position is not far from a double for the trailer.

Typically, the gammon price on 2 is much more important than the
gammon price on 1, because the cube usually will get turned, and
at that point it is the gammon prices on 2 that matter, even if
the cube is not taken.  The gammon price on 1 is significant
primarily for deciding whether or not to play for a gammon.

Gammon prices can be used to help guide doubling and taking
decisions in match play as well as money play, but extra care
must be taken due to fact that the gammon prices change with
each cube turn.  When both players are far from victory, the
match gammon price is around the money play value of 0.50.

David Montgomery
monty on FIBS
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