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. Whenone 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 ones superiority over the other, one�sdisposition, their current scores, and so on.However, there is always a ground for fairly calculated settlements withno regards to such external factors.

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

2.Theorem of Fair Settlements

Player shave accustomed to calculate their chances of hitting checkers in terms of thenumber of rolls out of 36, rather than in terms of percentage values. The following theorem is established on thisground, although the correspondence with the percentage system is simple inmany cases.

2.1. Theorem:

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

P: the number of chances, out of 36, forwinning the game,

Q: the current cube value of thegame.�

S: the fair settlement value of the game.

Ifthere 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 increasein a way based on the scale of the gammon potential. Section 7, Gammon Possibilities, discusses this pointfurther. From now on, no gammon isassumed.

In theory, if 36 games identical to the game in question are played, then P games would bewon 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 winand loss points would then be:

P*Q - (36-P)*Q

i.e. 2P*Q - 36Q

Therefore, theaverage settlement value of a single game would be:

S = (2P*Q - 36Q) / 36

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

Whether S ispositive or negative depends, of course, on the winning side. That would then indicate whether P-18 ispositive or negative, or equivalently whether P is greater or less than18.

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

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

After as implification by 18, the equivalent formula would Take its final form:

S = (1 - P/18) * Q

2.3.Illustration:

Considera 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. InitialSettlement Table

To produce asettlement table, values for P and Q must be given.�

3.1. P Values

P values runfrom 0 to 36. The 0 and 36 values mustobviously be eliminated because of the perfect winning certainty. Also, the P values from 19 to 35 essential lygenerate settlements equal to those of the opponent player from 17 to 1,respectively. Therefore, the two seriesof 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 todraw a single settlement table, only one cube value must be considered.� Furthermore, an easy table must allow theplayers 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 arepowers of 2. The number 16 has aspecial 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 extractionof 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 ormultiplying the answer by 2, once or twice.� This can hopefully be done on the fly, whileplaying.

So with the Qvalue set to 16 and the P values running from 1 to 17, the settlement tablelooks as follows:

4. RefinedSettlement Table

Inthis section, we�ll develop a settlement table that would be easy toremember.�

The obvious thing to start with is to Round the S values to their closest integralvalues.� Doing that would turn fractional numbers such as 15.11, 9.77 and 4.44, for examples, to whole numbers15, 10 and 4, respectively.�

Thisapproximation carries an error percentage with every whole number.� The error percentage is calculated asfollows, where S_a denotes the adjusted settlement:

Absolute value of (S_a S) / S_a.

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

Byinspecting column # 4, we notice that the errors fall in two ranges: 0.00% to6.66%, and 11.00% to 12.00%. Whereasthe first range is quite reasonable, the second one carries some surplus withit.

Noticethe following facts about the error column:

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

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

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

This should make us feel comfortable with the settlement values in the S_acolumn.�

Now let's drop from the table the columns carrying the fractional settlements anderrors, keeping only the columns with whole numbers.�

The final form of the settlement table that we have been working on now looks asfollows (For convenience, the S_a title hasbeen renamed S.):

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 developrules that have the merit to help players memorize the settlement numbers. The rules will be drawn from a series of observations.

Observation#1

Noticethat while P values increase from 1 to 17, S values decrease from15 to 1. This observation leads to thefoundation rule:

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

Observation#2

Thetable has a central line. It�s where Pand S carry their mid-values of 9 and 8 respectively. Try to remember this line.

Inthe 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 awinning chance of 25%, the fair settlement is 50% of the cube value.

Asground 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

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 settlementvalue of 12; also, the P values of 13 and 14 share the settlement value of4.

Inthe 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 cubevalue.

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 listtwo 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 sectiona, to 17-P in section b, and to 18-P in sectionc. This subtraction gives way to aneasy method to reach settlements for all P values from 1 to 17. This observation leads to rules for the mostfrequent 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 (HighSection Rule): For more than 13 good rolls, subtract that number from 18.

Forthe reader�s convenience, the settlement table and the settlement rules areseparately collected at the end of the article.

6.Applications

Exercise1:

Youand 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, hisside.� He offers to settle. What's a fair settlement figure?

Answer 1:

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 of16, the answer would be half of 12, namely 6.

Exercise2:

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

Answer 2:

Withtwelve 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.

Exercise3:

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

Answer 3:

On the fly, we figure out that 40% out of 36 is somewhere between 14 and 15. Say14.5. Remember that 14.5 is in theupper quarter.� The High-Section Ruledetermines S as 18-13.5, i.e. 3.5. Since the number 64 is 4 times the number 16 then multiply 3.5 by 2twice. 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 aswell 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 considerationwhether a gammon could occur in case of a miss. If it does, the amount of loss increases remarkably, and thereforethe 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 gammonpossibility 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 abovediscussion.

The idea will take us back to the proof of the Fair Settlement theorem. Theoretically speaking, if 36 identicalgames are played, then P games would win the face value of the cube each, and36-P games would loose double the face value each. The theoretical total point result would then be:

P*Q �(36-P)*Q*2

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

3PQ - 72Q

Whenwe 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 inthe case of no gammon the winning frequency was only half the games.) In the long run, the underdog gets paid onlyone third of such games, namely from P = 25 to P = 36.

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

S =(24Q-PQ)/12

Bysetting the cube value to 16, like we did in the no-gammon section, we concludethe formula:

S = 32 - 4P/3= 32 - 1.33 P

(Comparethis formula to the previous formula, S = 16 - 0.89 P.)