Fair Settlement

In Backgammon

## Play

1. Introduction

Occasionally, backgammon players run into situations when a settlement is desired toward the end of a chouette or money game.  When one or two more rolls would determine the fate of a game with a high cube value, players prefer sometimes to settle in order to cut their loss instead of riding a large risk.

Judges in situations can also seek settlements when conflict arises between two players.

Settlement agreements can be affected by non-mathematical factors associated with the players’ characters such as one’s superiority over the other, one’s disposition, their current scores, and so on.  However, there is always a ground for fairly calculated settlements with no regards to such external factors.

The study in this paper aims to determine a mathematical formula, and consequently a table, for settlements that seek mutual fairness to both players, which is based only on the assets of the game being played.

2. Theorem of Fair Settlements

Players have accustomed to calculate their chances of hitting checkers in terms of the number of rolls out of 36, rather than in terms of percentage values.  The following theorem is established on this ground, although the correspondence with the percentage system is simple in many cases.

2.1. Theorem:

In a backgammon game, let P, B and S refer to the following entities:

P: the number of chances, out of 36, for winning the game,

Q: the current cube value of the game.

S: the fair settlement value of the game.

If there is no chance for a gammon, then S is determined by the following formula:

S = (1 - P/18) * Q.

2.2. Proof:

Notice, first, that the theorem requires a no gammon chance, be it a gain or a loss.  If so, the settlement value would increase in a way based on the scale of the gammon potential.  Section 7, Gammon Possibilities, discusses this point further.  From now on, no gammon is assumed.

In theory, if 36 games identical to the game in question are played, then P games would be won and 36-P games would be lost.  The amount of points won would be P*Q, and the amount of points lost would be (36–P)*Q.  In total, the combined win and loss points would then be:

P*Q - (36–P)*Q

i.e.        2P*Q – 36Q

Therefore, the average settlement value of a single game would be:

S = (2P*Q – 36Q) / 36

i.e.        S = [(P-18)/18] * Q

Whether S is positive or negative depends, of course, on the winning side.  That would then indicate whether P-18 is positive or negative, or equivalently whether P is greater or less than 18.

In order to produce a positive settlement value, let P be less than 18.  Then:

S = [(18-P)/18] * Q.

After a simplification by 18, the equivalent formula would take its final form:

S = (1 – P/18) * Q

2.3. Illustration:

Consider a game where the cube is on 64 and where there are 5 good rolls to win:

P = 5; Q = 64.  ==>

S = [(18 - 5) / 18] * 64 = 46.22

3. Initial Settlement Table

To produce a settlement table, values for P and Q must be given.

3.1. P Values

P values run from 0 to 36.  The 0 and 36 values must obviously be eliminated because of the perfect winning certainty.  Also, the P values from 19 to 35 essentially generate settlements equal to those of the opponent player from 17 to 1, respectively.  Therefore, the two series of settlement values differ in their sign only.

As a result, the P values worth being considered are those from 1 to 17 only.

3.2. Q Values

In order to draw a single settlement table, only one cube value must be considered.  Furthermore, an easy table must allow the players to extract settlements for other desired cube values.

The cube value of 16 seems to be a convenient choice.  Let’s explain why.

Q values are powers of 2.  The number 16 has a special place among frequent cube values when settlements are desired.  As the fourth power of 2, the number 16 stands in the middle of a series of cube values 4, 8, 16, 32, and 64.  This particularity allows an easy extraction of settlement values for the other four cube values.  The reason is that the settlement for any desired one of these cube values could be extracted from the table by easily dividing or multiplying the answer by 2, once or twice.  This can hopefully be done on the fly, while playing.

So with the Q value set to 16 and the P values running from 1 to 17, the settlement table looks as follows:

 P S 1 15.11 2 14.22 3 13.33 4 12.44 5 11.55 6 10.66 7 9.77 8 8.88 9 8.00 10 7.11 11 6.22 12 5.33 13 4.44 14 3.55 15 2.66 16 1.77 17 0.88

4. Refined Settlement Table

In this section, we’ll develop a settlement table that would be easy to remember.

The obvious thing to start with is to round the S values to their closest integral values.  Doing that would turn fractional numbers such as 15.11, 9.77 and 4.44, for examples, to whole numbers 15, 10 and 4, respectively.

This approximation carries an error percentage with every whole number.  The error percentage is calculated as follows, where Sa denotes the adjusted settlement:

Absolute value of (Sa – S) / Sa.

The next form of the table adds to the previous one the new settlement numbers and their corresponding error percentages:

 P S Sa Error 1 15.11 15 0.73% 2 14.22 14 1.57% 3 13.33 13 2.53% 4 12.44 12 3.66% 5 11.55 12 4.09% 6 10.66 11 3.09% 7 9.77 10 2.30% 8 8.88 9 1.33% 9 8.00 8 0.00% 10 7.11 7 1.38% 11 6.22 6 3.67% 12 5.33 5 6.66% 13 4.44 4 11.00% 14 3.55 4 11.25% 15 2.66 3 11.33% 16 1.77 2 11.50% 17 0.88 1 12.00%

By inspecting column # 4, we notice that the errors fall in two ranges: 0.00% to 6.66%, and 11.00% to 12.00%.  Whereas the first range is quite reasonable, the second one carries some surplus with it.

Notice the following facts about the error column:

1. The top twelve numbers fall within the low 6.66% error.  This is a very good approximation.

1. Only five values fall in the high 11%-to-12% error range.

When an accuracy of 11%-to-12% is not close enough, players must remember that the P values involved are those highest five, from 13 to 17.

This should make us feel comfortable with the settlement values in the Sa column.

Now let’s drop from the table the columns carrying the fractional settlements and errors, keeping only the columns with whole numbers.

The final form of the settlement table that we have been working on now looks as follows (For convenience, the Sa title has been renamed S.):

 P S 1 15 2 14 3 13 4 12 5 12 6 11 7 10 8 9 9 8 10 7 11 6 12 5 13 4 14 4 15 3 16 2 17 1

# 5. Settlement Rules

The table is good enough as a reference, but it may not be simple enough to know by heart.  In this section, we will develop rules that have the merit to help players memorize the settlement numbers.  The rules will be drawn from a series of observations.

Observation #1

Notice that while P values increase from 1 to 17, S values decrease from 15 to 1.  This observation leads to the foundation rule:

Rule #1 (Foundation Rule): The lower the chances, the higher the settlement.

Observation #2

The table has a central line.  It’s where P and S carry their mid-values of 9 and 8 respectively.  Try to remember this line.

In the game, the central line simply means that when your chance of winning is 9 (out of 36) then the settlement value would be exactly 8.  If you recall that the table’s default cube value is 16, then the central line translates into the second rule:

Rule #2 (Basic Rule): For a winning chance of 25%, the fair settlement is 50% of the cube value.

As ground for the next two observations, imagine the 17 rows partitioned into three sections: First 4, then middle 9, then last 4.  We’ll call this partition the 4-9-4 partition (See the highlighted figure below):

1. Rows 1 to 4
2. Rows 5 to 13
3. Rows 14 to 17

 P S 1 15 2 14 3 13 4 12 5 12 6 11 7 10 8 9 9 8 10 7 11 6 12 5 13 4 14 4 15 3 16 2 17 1

To remember this partition, just bear in mind the numbers 5 and 13.

Observation #3

Notice that S repeats the same value as it moves from one section to the next.  The P values of 4 and 5 share the settlement value of 12; also, the P values of 13 and 14 share the settlement value of 4.

In the game, this means that having 4 or 5 good rolls to win yields the same 75% settlement of the cube value.  Similarly, 13 or 14 good rolls yield the same 25% settlement of the cube value.

If we notice that 4 or 5 rolls actually translate to a 12.5% chance (one eighth), and that 13 or 14 rolls translate to 37.5% (three eighths), then we can list two new rules:

Rule #3: For a winning chance of 12.5%, the fair settlement is 75%. (6 times the cube value)

Rule #4: For a winning chance of 37.5%, the fair settlement is 25%. (Only 2/3 of the cube value)

Observation #4

Notice, in the 4-9-4 partition above, that S is equal to 16-P in section a, to 17-P in section b, and to 18-P in section c.  This subtraction gives way to an easy method to reach settlements for all P values from 1 to 17.  This observation leads to rules for the most frequent situations:

Rule #5 (Low-Section Rule): For less than 5 good rolls, subtract that number from 16.

Rule #6 (Mid-Section Rule): For 5 to 13 good rolls, subtract that number from 17.

Rule #7 (High Section Rule): For more than 13 good rolls, subtract that number from 18.

For the reader’s convenience, the settlement table and the settlement rules are separately collected at the end of the article.

6. Applications

Exercise 1:

You and your opponent are bearing off.  You have 2 checkers on point 1 and he has 3 checkers on point 2.  It’s his roll and the cube is on 8, his side.  He offers to settle.  What’s a fair settlement figure?

The opponent’s chance of winning is 5 out of 36 (any set but 1-1).  Because 5 is in the middle section then the settlement is 17-5= 12 (Mid-Section Rule).  Knowing that the cube value of 8 is half of the default cube value of 16, the answer would be half of 12, namely 6.

Exercise 2:

You figure out that any roll of 2 (direct or combined) would give you a win in a game with a cube value of 32 on your side.  You desire to settle.  What would be a fair settlement?

With twelve chances to roll 2 (any 2, or 1-1), P is equal to 12.  Since 12 is in mid-section, S will be 17-12 = 5.  Since the cube is at 32, a fair settlement would be to pay 10 points.

Exercise 3:

What’s a fair settlement for a 40% chance of winning a game with a cube on 64?

On the fly, we figure out that 40% out of 36 is somewhere between 14 and 15. Say 14.5.    Remember that 14.5 is in the upper quarter.  The High-Section Rule determines S as 18-13.5, i.e. 3.5.  Since the number 64 is 4 times the number 16 then multiply 3.5 by 2 twice.  The answer is to give 14.

Recall that the upper P quarter has an 11%-to-12% error range.  This means the settlement of 14 could as well be any number from 12 to 16.

7. Gammon Possibilities

This point is very important to bear in mind before using the settlement table.  A settlement must take into consideration whether a gammon could occur in case of a miss.  If it does, the amount of loss increases remarkably, and therefore the formula generated above should not be applied.

Is it possible to generate a formula for the case of a gammon possibility?

The answer depends on whether more information can be provided.  We need to know the percentage of gammon possibility in case of a miss.  Of course, building a formula for all possibilities is beyond our purposes, but it is possible to consider one special case.  That case is when a 100% gammon loss would occur if a miss occurs.  We must indicate that such a situation is infrequent, but we will discuss it in order to draw a comparison with the above discussion.

The idea will take us back to the proof of the Fair Settlement theorem.  Theoretically speaking, if 36 identical games are played, then P games would win the face value of the cube each, and 36-P games would loose double the face value each.  The theoretical total point result would then be:

P*Q – (36-P)*Q*2

(Notice the multiplication by 2 for the loss.)  By simplifying the expression, the total becomes:

3PQ – 72Q

When we divide by 36, the theoretical value of a single game would then be equal to:

S = (PQ- 24Q)/12

We notice that S becomes zero when P is 24.  So, a tie occurs when the underdog player holds 24 out of the 36 cards, namely two thirds.  (Remember that in the case of no gammon the winning frequency was only half the games.)  In the long run, the underdog gets paid only one third of such games, namely from P = 25 to P = 36.

Like we did in the case of no gammon, if we take P in the lower segment, i.e. less than 24, then the formula would be:

S = (24Q-PQ)/12

By setting the cube value to 16, like we did in the no-gammon section, we conclude the formula:

S = 32 – 4P/3 = 32 – 1.33 P

(Compare this formula to the previous formula, S = 16 – 0.89 P.)

The table for the new formula is not as attractive as that of the no-gammon case.  However, it would give a clear idea if we draw the curves of the two formulas on the same graph.  The next graph shows how the gammon and no-gammon formulas would compare to each other. As you can see, the settlement price gets much higher when a gammon is possible.

# 8. Conclusion

The study generates a mathematical formula to calculate fair settlements for chouette games.  The formula is based on two input items, the cube value and the chance of winning the game.  From the formula, an easy-to-remember table was drawn, and a set of rules was derived.

The focus of the study is on the assumption of no gammon.  However, a section on the possibility of gammon is presented at the end.

 P S 1 15 2 14 3 13 4 12 5 12 6 11 7 10 8 9 9 8 10 7 11 6 12 5 13 4 14 4 15 3 16 2 17 1

Settlement Table

for the 16-cube value

P: Winning possibilities out of 36

S: Settlement value

Settlement Rules

Rule #1 (Foundation Rule):

The lower the chances, the higher the settlement.

Rule #2 (Basic Rule):

For a winning chance of 25%, the fair settlement is 50% of the cube value.

Rule #3:

For a winning chance of 12.5%, the fair settlement is 75%. (6 times the cube value)

Rule #4:

For a winning chance of 37.5%, the fair settlement is 25%. (Only 2/3 of the cube value)

Rule #5 (Low-Section Rule):

For less than 5 good rolls, subtract that number from 16.

Rule #6 (Mid-Section Rule):

For 5 to 13 good rolls, subtract that number from 17.

Rule #7 (High Section Rule):

For more than 13 good rolls, subtract that number from 18.