Backgammon End Game Doubling Strategies in Money Play and Match Play

Backgammon End Game Doubling Strategies in
Money Play and Match Play
Christina Erceg
Department of Mathematics and Computer Science
Westminster College, Pennsylvania, USA
[email protected]
March 21, 2013
Abstract
In this paper we study end game situations and doubling strategy in Backgammon. Doubling is a key strategy in the game since it determines the worth of
the game and is a key factor for determining a winner. The paper begins with
an overview of how to play Backgammon. Next, we determine the expected
values for when to double and accept a double in money play, based on what
point in time we are in the game. Then, we look at doubling from a match play
perspective. Overall, we compare doubling decisions for end game situations in
money play to match play.
1
Introduction
Backgammon is a game full of wonderful mathematical calculations. According to
Backgammon History, originally written by Chuck Bower [2], Backgammon’s roots
date back to 3,000 BC during the Mesopotamian era. Similar versions of the game
have been played for over 5,000 years, making it the oldest played board game. In
medieval Europe the game was very popular, along with a version of the game in
the Indian and Egyptian cultures. The modern version of Backgammon began in
the 1920’s in New York, where an unknown gambler brought up ideas of many rules
including use of the doubling cube. In the 1960’s, the first tournaments of the game
began and were known as the World Championships. In the 1990’s the first online
versions of the game attracted many new players. Today, Backgammon continues to
be a widely played game among the people of Greece, Rome, and all across America.
For more information on the history of the game, refer to [2].
Backgammon is a two player board game that begins with the set-up in Figure 1.
In the game, each player uses their 15 checkers, the 24 pegs of the board, and two
1
Figure 1: Beginning board set-up
dice to move in opposite directions, toward their home board, which is the final six
pegs. After all of a player’s pieces are in their home board they begin a process called
bearing off, which entails using the rolls of the dice to move the pieces from a player’s
home board off the game board. Whichever player completes this first wins the game.
In Figure 1, the red player’s home board is pegs 1 through 6. The white player’s home
board is pegs 19 through 24. Let’s look at a couple examples of how the bearing off
process works.
Example 1: You have all your pieces in your home board. The next step is to get
your pieces off the board. Suppose you have a piece on the 5-point and a piece on the
2-point. Let’s say you roll a 2 and a 3. Moving your piece from the 2-point 2 places
moves it off the game board. This is as example of the definition of bearing off. Next
moving your piece from the 5-point 3 places moves it to the 2-point. Since the piece
is still located on the game board, it was not bore off. We can see that in order for
a piece to bear off the board, a number on a die or a combination of numbers from
the dice need to produce a number greater than or equal to the number of pegs away
from the edge of the board a piece is on.
Example 2: Suppose you have two checkers left in your home board during your
bearing off process, 1 on the 3-point and 1 on the 5-point. You roll a 2 and a 6. You
choose to move your piece on the 3-point to the 1-point, which is not bearing off but
getting your piece closer to getting off the board. Also, you move your piece from the
5-point off the game board with your roll of the 6, which is an example of bearing
off.
A doubling cube is also used in the game, which multiplies what the current worth of
the game is by two. Doubling continues to be active in the game even if a player has
already multiplied the stakes once. The process alternates between players for who
decides they want to double the stakes again. After a first double has been declared,
the subsequent doubles in a game are called redoubles. There are certain strategies
for doubling in the game, which complicates the strategy. We will discuss this further
in Section 2.2.
2
Backgammon can be played by individual games, called money play, or by matches,
called match play. Money play determines a winner based on a single game. We will
talk more about money play in Section 3. Match play determines a winner based on
a certain number of points and whichever player gets those points first wins. We will
discuss match play more in-depth in Section 4.
Near the end of the game, each player can calculate where they stand in the game
and whether or not they should double the stakes of the game. Previously known
findings include calculating the doubling point at the very beginning of a game [4]
and the last move of a game [1]. The doubling point is the prime point in the game
a player should be at before they choose to double. Also known are one and two
checker ending position guidelines depending on how far each player is from winning
in match play [5]. In this paper, we determine how to calculate the probabilities for
whether or not a player should double and if the other player should accept for both
money play and match play. We determine how decisions differ based on the type
of play. Depending on which type of play, the rules for doubling are different since
money play is based on one game, but match play is played to a certain point total.
We begin the paper in Section 2 by discussing how to play the game and some key
rules and definitions that are part of the game. Some key points of this section include
how the dice rolls affect the game, doubling, and beavers. Money play is discussed
in Section 3, including money equity, when to double, when to accept a double, end
game situations involving doubling, and the Jacoby Paradox. Match play is discussed
in Section 4, including match equity and end game situations involving doubling.
Section 5 compares doubling in end game situations in money play to doubling in end
game match play situations. We find that doubling decisions for money play differ
when compared to match play in borderline cube action cases. Section 6 provides
some concluding remarks and the direction for the next step in this research project.
2
How to Play
This section includes rules that pertain to the calculations used in this paper. Included
are basics of the dice involving legal moves in the game, the term doubling, and what
it means to beaver. For a complete list of rules of the game, refer to [3].
2.1
Basics of the Dice
When moving around the game board, a player uses the number on one die to move
that exact number of places to a new peg, followed by the same process with the
another die.
Example 3: Suppose you are just beginning a game. Referring to Figure 1, you
are the white pieces. Let’s say you roll a 2 and a 4. You may use your 2 to move
a piece from the 12-point to the 14-point. With your 4 on the other die, you may
choose to move a piece from the 12-point to the 16-point. A different possibility is
3
that you move a piece from the 17-point to the 19-point with the 2 and a piece from
the 17-point to the 21-point with the 4. These are just two of the many different
options that a player has to move.
If the numbers on both dice are the same, then you have rolled doubles. In the game,
if you roll doubles, then you can roll the amount of each die twice, meaning 4 times
the amount of one die.
Example 4: Again, suppose you are just beginning a game and that you are the
white pieces on the game board of Figure 1. Let’s say you roll double 1’s. Then,
you can choose to move 4 different pieces 1 space each, 1 piece 4 spaces, or another
similar combination. For example, let’s say you moved a piece from the 17-point to
the 18-point, and then again from the 18-point to the 19-point. For another piece
starting on the 17-point, suppose you complete the same series of moves as the first
piece.
One legal move in a game is moving to an open peg, as in the moves in Example 3.
Another legal move is to move to a peg containing your pieces, such as the player’s
choices in Example 4. A third legal move is to move to a peg that contains a single
piece from your opponent. In this instance, your opponent would have to move his
piece back to the beginning. To visualize this legal move, let’s look at the following
example.
Example 5: Suppose you have 3 pieces on your 14-point and your opponent has a
single piece on the 16-point, which is in your path. You roll a 2. Then, it is legal for
you to move a piece from your 14-point to the 16-point to “hit” your opponent.
It is important to note that in the game, it is illegal to move to a peg that contains
two or more of your opponent’s pieces.
Example 6: Suppose you have a piece on the 14-point and your opponent has 3
pieces on the 16-point. If you roll a 2, then it is illegal to move your piece from the
14-point to the 16-point because your opponent has more than two pieces on that
point.
2.2
Doubling
At the start of each game, the game is worth 1 point. In most games, if a player feels
that they have a large enough advantage over their opponent, then they may wish to
double the stakes by using the doubling cube. The player can only double if it is his
turn and he has not yet rolled the dice.
If a player doubles, then their opponent has two options. The first is that they can
accept the double and continue playing the game. In this instance, if they feel as if
they have a better lead, then they can redouble at any time to make the game worth
4 points. This cycle of redoubling has no limit, but it alternates between the players
for whose decision it is to double. The alternating of the cube is referred to as having
possession of the cube, or owning the cube. Secondly, if the player does not feel as if
4
his chances of winning are high enough, he can decline the double. In this instance,
the game is over, and the player who proposed the double gets the current amount
of points that were in play for the game. If you are playing a match up to n points
and one player has n-1 points, for the first time going into a new game, the doubling
cube is not allowed during that game. This is called the Crawford Rule. [6]
2.3
Beaver
To beaver is a special type of doubling rule. It is when a player doubles, yet the
other player feels as if their position is good enough to redouble. They may do so
immediately. This is done without giving up the cube, meaning that the player who
redoubled still owns the cube.
3
Money Play
One way to play Backgammon is called money play. The game is won or lost based
on one game. With this being said, it is important to look at probabilities of winning
to determine what move should be made. If you lose the game, then you have lost
overall.
In this section, Table 1 shows all of the possible roll combinations there are in a
game. Then we calculate money equity, which helps determine which player has an
advantage in the game. Both of these concepts are useful for the next part of the
section which uses money equity to determine if the player on roll should double and
if their opponent should accept or decline. Lastly, we determine that a probability
greater than .5 is the doubling point for a last move situation and a probability of .8
is the doubling point for the very beginning of the game.
3.1
Probabilities of Roll Combinations
To further study end game situations, we need to know which roll combinations we
need to consider during our calculations. Table 1 is formed by considering the 36
possible roll combinations and which dice combinations will yield the desired output
of bearing off.
5
Table 1: Chances to bear off one or two men with one roll
a piece on the
off of board
1 pt
2 pt
3 pt
4 pt
5 pt
6 pt
off of board
off of board
36
36
36
34
31
27
1 pt
36
36
36
34
29
23
15
2 pt
36
36
26
25
23
19
13
3 pt
36
34
25
17
17
14
10
4 pt
34
29
23
17
11
10
8
5 pt
31
23
19
14
10
6
6
6 pt
27
15
13
10
8
6
4
Example 7: Suppose player 1 has a piece on the 2-point and a piece on the 4-point
and he wants to know how many roll combinations will bear his pieces off on the next
roll. Looking at the 2-point on the y-axis and the 4-point on the x-axis we see that
these points intersect at 23. So player 1 has 23
chance of bearing off his pieces on the
36
current roll.
Table 2 is a similar version of Table 1, except using probabilities which will be easier
to use in computations.
Table 2: Table 1 translated to probabilities
a piece on the
off of board
1 pt
2 pt
3 pt
4 pt
5 pt
6 pt
3.2
off of board
off of board
1
1
1
.94
.86
.75
1 pt
1
1
1
.94
.81
.64
.42
2 pt
1
1
.72
.69
.64
.53
.36
3 pt
1
.94
.69
.47
.47
.39
.28
4 pt
.94
.81
.64
.47
.31
.28
.22
5 pt
.86
.64
.53
.39
.28
.17
.17
6 pt
.75
.42
.36
.28
.22
.17
.11
Money Equity
Let X be the value amount that you get from the game. Then money equity is defined
to be E(X). Let’s look at an example of calculating money equity.
Example 8: Suppose the last two pieces that player 1 has are on the 2-point and
the 5-point and player 2 has two pieces on the 1-point. There are no doubles in the
game, so the game is worth one point. Using Table 1, we can see where the 2-point
and the 5-point intersect to find player 1’s chance of getting off the board on his next
roll. Then we can calculate the money equity of player 1 which is 19
(1) + 17
(-1) =
36
36
2
.
36
6
Money equity is calculated to determine which decision a player should make when it
comes to a doubling decision. A higher equity of doubling compared to not doubling
for a player means the player should double. Likewise, a higher equity of accepting a
doubling compared to declining the double means a player should accept the double.
3.3
End Game Situation
Let’s look at an end game situation that uses the concept of money equity to determine
if a player should double and if their opponent should accept.
Example 9: Suppose player 1 has a piece on the 5-point and a piece on the 2-point
and player 2 has a two pieces on the 1-point. It is player 1’s turn to roll and there
have been no doubles in the game thus far. Let’s examine the following questions:
1. What is player 1’s chance of winning?
2. Should player 1 double?
3. If player 1 doubles, should player 2 accept?
Solution:
First let’s discuss the question of player 1’s chance of winning. We know that player
1 wins only if he is able to bear off his two men on the next roll since any combination
of the dice will give player 2 a win. The tables above are useful for they give the
chances of bearing off with players on one or two of the last six pegs. We can easily
see by reading Table 2, that player 1 has a 53% chance of winning.
Second, let’s look at whether or not player 1 should double. Remember that Table
possibility of bearing off and winning the game.
1 showed that player 1 has a 19
36
17
Therefore, there is a 36
chance that player 1 loses since player 2 will always win on
their next turn. Looking at the expected values of winning for player 1, we have if
2
(1) + 17
(−1) = 36
. If player 1 doubles, then
player 1 does not double, E(X) = 19
36
36
19
17
4
E(X) = 36 (2) + 36 (−2) = 36 . Player 1 doubling produces a higher equity which
means that player 1 should double so let’s assume that’s what the decision is.
Now we have two situations, either player 2 will accept the double and continue to play
or player 2 will decline the double and lose the game. If player 2 accepts, then the game
is now worth 2 points. We have that player 1 still wins 19
of the time and loses 17
of the
36
36
17
−4
time. Looking at expected values, if player 2 accepts, E(X) = 36
(2) + 19
(−2)
=
.
36
36
Our second situation is if player 2 declines the double. We have that the expected
value of winning is −1. In this situation, player 1 would automatically win the game
and the 1 point that was in play. For player 2, it is worth the risk to accept the double
and hope that player 1 does not bear off the board with one roll. If this occurs, then
player 2 would win 2 points by taking one roll of the dice on his next turn.
7
As you can see, some of the most important decisions come in the last few moves
of a game. The following proposition gives a general idea of a last move situation
involving doubling.
Proposition: Suppose it is Player 1’s roll. If there is 1 move left in the game then
1. Player 1 should double if his chance of winning is greater than .5.
2. Player 2 should accept the double when his chance of winning is greater than
.25.
Proof: Let p be the probability of winning the game and let x be the current worth of
the game. The first expected value that we look at is one without a double in the game.
We have p(x) + (1 − p)(−x) = p(x) − (x) + (px) = 2px − x. The next expected value is
when a double was accepted. So, p(2x) + (1 − p)(−2x) = 2px − 2x + 2px = 4px − 2x.
We will then compare these two equations to see when the equity of doubling is greater
than the equity when there wasn’t a double. Therefore, we have 2px − x ≤ 4px − 2x,
which means x ≤ 2px. So, 12 ≤ p. If the probability is greater than .5, then player 1
should double. Assume player 1 has doubled. Calculating money equity, if player 2
accepts the double then we have (p)(2x) + (1 − p)(−2x) = 2px − 2x + 2px = 4px − 2x.
The expected value of winning if player 2 declines the double is −x. We want to see
when the probability of accepting the double is greater so we have 4px − 2x ≥ −x,
which means 4px ≥ x. Then, p ≥ 41 . Therefore, player 2 should accept the double
when his chance of winning is greater than .25.
3.4
Doubling at the Start of a Game and in a Last Move
Situation
In section we will determine that when a player’s probability of winning, p, is greater
than or equal to .8 then a player should double at the beginning of the game. In
order to do this we will introduce Lemma 1, Lemma 2, and Theorem 3. Lemma 1
uses the probability that player 1 will win to determine the probability that an event
E happens. Lemma 2 shows that player 1’s doubling point is player 2’s folding point.
Theorem 3 uses Lemma 1 and Lemma 2 to determine player 1’s doubling point. Using
the previous proposition, we can see that doubling points are different throughout the
game since each move made in the game can change, sometimes drastically, what each
player’s probability of winning the game is.
3.4.1
Beginning of the Game
First, we will look at when to double at the beginning of the game. To better analyze
the situation, suppose we are looking at a continuous game, meaning that the probability of winning is instantaneously changing. Lemma 1, Lemma 2 and Theorem 3
were proved in [4].
8
Lemma 1 Let x be the current probability of winning for player 1. If E is an event
then the probability that we get to x + b, player 1’s doubling point, before x − a, player
a
2’s doubling point is P(E) = a+b
.
Proof: Let x be the probability that player 1 will win. Then,
x = P (E)P (x | E) + P (∼ E)P (x | −E) = P (E)(x + b) + P (−E)(x − a),
where −E is the event that E does not happen.
So we have
x = P (E)(x + b) + (1 − P (E))(x − a).
Therefore,
P(E) =
a
.
a+b
Lemma 2 Assume only player 1 has the right to double and that his probability of
winning the game is α. Let α0 be player 1’s doubling point and let β 0 be the point at
which player 2 folds. Then, α0 = β 0 .
Proof: If player 2 would fold, then player 1 should double. This gives us that α0 ≤
β 0 . Suppose there exists γ such that α0 < γ < β 0 . Let’s look at the difference when
it comes to doubling at α0 verses doubling at γ. If player 1 wins, then it doesn’t
matter because he will receive all the points. If however, player 1 does not win, then
it is better to double at γ. Therefore, it is too early to double at α0 and α0 ≥ β 0 .
We can use these lemmas to find the probability for doubling very early in the game
in our next theorem.
Theorem 3 Player 1’s doubling point, α0 = .8.
Proof: Let s be the current value of the game. If player 2 declines the double, then
the money equity of player 2 is −s. We know by symmetry that player 2 doubles
at 1-α0 , and at this point player 1 would decline the double. Lemma 2 tells us that
player 1 will win before player 2 doubles at a probability of (1-α0 )/α0 . With all this
information, we can set player 2’s money equity from declining the double to the
money equity using player 1’s doubling point.
−s = 2s((1 − α0 )/α0 − (2α0 − 1)/α0 )
After simplifying this expression, α0 = .8. This means that player 1 should wait
to double until his chance of winning is 80%. Thinking about this probability, it is
very early in the game so doubling at too low of a probability has a higher chance
of hurting the player who doubled, near the end of the game. A higher probability
for doubling at the beginning of the game is due to the fact that there are so many
possible outcomes that the game can have. Just because a player is currently ahead
doesn’t necessarily mean they will stay ahead the whole game.
9
3.5
Jacoby Paradox
The Jacoby Paradox states that in certain situations it is not wise for a player to
redouble, however an improvement in their opponent’s position can make it wise to
redouble when the player on roll owns the cube and has one last chance to choose to
redouble. This can best be summed up through an example.
Example 10: Suppose player 1 has pieces on the 2-point and the 5-point and player
2 has a piece on the 6-point. Both players are in the process of bearing off and player
on the next roll and currently
1 owns the cube. Player 1’s chance of winning is 19
36
the game is worth 2 points. Should player 1 redouble?
Solution: Let’s look at the different possibilities that can occur by examining the
expected values of player 1 winning the game.
If player 1 does not redouble, then player 1’s equity is:
.53(2) + .25(.47)(2) + .35(−2) = .6
This money equity is found from calculating the probability that player 1 gets off on
his next roll. Then if they do not we use the probability that player 2 does or does
not get off. Notice that the value of the game in this case is 2.
If player 1 redoubles and player 2 accepts, then player 1’s equity is:
.53(4) + .47[.75(−8) + .25(8)] = .24
In this case, the value of the game is worth 4 if player 1 gets off right away, or worth
8 if he does not because player 2 would always redouble the game again.
If player 1 redoubles and player 2 declines, then player 1’s equity is: −1
Assume player 1 did not roll high enough to bear off completely and it is now player
2’s turn and he rolled a 1 and a 2. Player 2’s position has improved in the game from
a piece on the 6-point to a piece on the 3-point.
If player 1 doesn’t redouble, then player 1’s equity is:
.53(2) + .47(−2) = .12
Player 1’s money equity differs from above since now player 2 will bear off on their
next roll no matter what he rolls.
If player 1 redoubles and player 2 accepts, then player 1’s equity is:
.53(4) − .47(4) = .24
What the paradox tells us is that initially player 1 should not redouble to make the
game worth 4 points because his equity would be less than if he chose not to redouble.
However, an improvement in player 2’s position in the game makes it plausible for
player 1 to now double, which is shown by the higher equity if he redoubles.
10
4
Match Play
Match Play is another way to play Backgammon. It is based on a series of games
called a match, whose number of points the match is played to is predetermined. Most
occurrences play with a match to five or seven points. If you lose a game, it does not
mean the match is necessarily over. When playing match play it is important to base
decisions on how many points each player has overall.
In this section we determine how to calculate match equity, which is useful for determining whether a match will be won or lost. A table of match equities is included
and is based on how many points away each player is from winning the match. We
then look at an end game situation that uses match equity to determine if a player
should double and if their opponent should accept or decline.
4.1
Match Equity
Sometimes when keeping track of you and your opponent’s standings in a match play
game, the term “away” might be used. It is common among Backgammon players.
We say that the game is n-away, m-away where n and m describe how many points
each player is from winning the match. For example, a match may be 2-away, 3-away.
What it means is that one player is 2 points away from winning the match and the
other player is 3 points away from winning the match. This term can be broadened
to other amounts of points away from winning the game.
Example 11: Suppose the score of a match is 4-2, played to 5 points. Then, the
player who has 4 points is said to be 1-away and the player with 2 points is said to
be 3-away.
We define match equity to be the probability of winning the match. Based on how
many points away each player is from winning the match, we can find what probabilities result for winning for each player. Table 3 gives the match equity for both
players in the match. On the y-axis is how many points player 1 is from winning the
match, who is the player we are determining the equity for. On the x-axis is how
many points player 2 is from winning the match, who is the opponent.
11
Table 3: Match Equity Table
1
2
3
4
5
1
.50
.30
.25
.17
.15
2
.70
.50
.40
.32
.25
3
.75
.60
.50
.41
.34
4
.83
.68
.59
.50
.42
5
.85
.75
.66
.58
.50
[7]
The following example uses the chart to determine the equity for each player.
Example 12: Suppose you are leading 3 to 1 in a 5 point match. So, you have 2
points to go, making you 2-away. Your opponent has 1 point, so they are 4-away. We
look at 2 on the y-axis and 4 on the x-axis to see where the two numbers intersect.
From this, we find that the player who is 2-away has an equity of .68 and the player
who is 4-away in turn has 1 − .68 = .32 equity. This makes sense since we know that
both players equities when added together will equal one, since we are certain one of
the players will win the match.
4.2
End Game Situations
There are two methods, one using money equity and one using match equity, that
are used to figure out whether or not a player should double and if their opponent
should accept in match play. In the following example, we will analyze a situation
using both of the methods to show that they are equivalent.
Example 13: Say player 1 is 2-away from winning the match and player 2 is 3away from winning the match. Assume player 1 doubles and it is player 2’s decision
to accept or decline. We will calculate the match equity of player 2 to determine
which decision produces a higher equity.
First, we will complete the calculations using match equity.
If player 2 declines, then the score would be 1-away, 3-away, so player 2’s match equity
is .25.
If player 2 accepts and wins, then the score would be 2-away, 1-away, so player 2’s
match equity is .70.
If player 2 accepts and loses, then player 2’s match equity is 0.
Setting the decline and accept amounts equal to each other,
.25 = p(.75) + (1 − p)(0)
1
=p
3
Now we complete the same process using money equity.
12
If player 2 declines, then the equity is .5.
If player 2 accepts and wins, then the equity is .25.
If player 2 accepts and loses, then the equity is -1.
Setting the accept and decline amounts equal to each other,
.5 = p(1) + (1 − p)(.25)
.5 = p + .25 − .25p
.5 = .75p + .25
.25 = .75p
1
=p
3
As we can see, both methods produce the result of 13 .
Our next lemma is another conclusion that involves both player 1 and player 2’s
match equities.
Lemma 4 Player 2’s match equity of winning = 1 - Player 1’s match equity of winning
Proof: We have that player 1’s match equity of winning + player 2’s match equity
of winning = 1. This is true since the equity of winning will always be 1, meaning
the game will always be won by some player. So, player 2’s match equity of winning
= 1 - player 1’s match equity of winning.
Next, we will use these following facts to complete an end game situation involving
2-away, 3-away and known positions of pieces for both player 1 and player 2.
Example 14: Suppose player 1 has a piece on the 2-point and a piece on the
5-point. Player 2 has two pieces on the 1-point. It is player 1’s turn to roll. Should
player 1 double? If so, Should player 2 accept?
Solution: First we must figure out if player 1 should double. Let’s assume that player
1 doesn’t double to start. Then the equity of player 1 can be calculated by the chance
that player 1 bears off on the next roll times what the match equity would be if he
won that point. This is added to the chance that player 1 does not bear off on his
next roll multiplied by the match equity that would occur if player 2 got the point.
This is summarized in Table 3.
Table 3: Equity of Player 1-No Double
win
lose
19
36
17
36
.75 (since 1-away, 3-away)
.50 (since 2-away, 2-away)
We can use this table to calculate the match equity of player 1 if he doesn’t double.
So we have that:
19
17
(.75) + (.50) = .64
36
36
13
Using a similar idea we see that if player 1 doubles, then player 1’s equity is:
Table 4: Equity of Player 1-Doubles
win
lose
19
36
17
36
1 (match over)
.30 (since 2-away, 1-away)
Then, we have that
19
17
(1) + (.3) = .67
36
36
Comparing the equity of player 1 if they do not double and the equity of player 1 if
they do double, we can see that if player 1 doubles then they have a higher equity.
Therefore, player 1 should double.
Next, we determine if player 2 should accept player 1’s double.
If player 2 declines the double, then the equity of player 2 is .25 since it would be
1-away, 3-away.
If player 2 accepts the double, then the equity of player 2 is:
Table 5: Equity of Player 2-Accepts Double
win
lose
17
36
19
36
.70 (since 2-away, 1-away)
0 (match over)
Then, we have that
19
17
(.7) + (0) = .33
36
36
Comparing the equity of player 2 when they decline the double compared to when
they accept the double, we can see that if player 2 accepts the double then they have
a higher equity. Therefore, player 2 should accept the double.
We can use the information from player 2 of whether to accept or decline the double
to form a more general principle.
5
Doubling in End Game Situations Comparisons
In this section we compare similarities and differences that are found for doubling in
end game situations for money play versus match play. First, we will look at a chart
analyzing doubling decisions for end game money play. Then we will look at similar
charts except for match play 2-away, 3-away and 2-away, 4-away. With these charts
we spot where the similarities and differences lie in the decision making process for
doubling.
14
5.1
Money Play Doubling Decisions
The following result was discovered by Bob Koca in the article, Curing Your Short
Bearoff Blues [5]. For end game situations there are 5 cube actions that can be
completed during the bearing off process. For the purposes of Table 5 we will define
the cube actions as follows:
1. Not good enough to double/beaver if doubled
2. Not good enough to double/not a beaver if doubled
3. Good enough for an initial double but not a redouble/take if doubled
4. Good enough to redouble/take
5. Good enough to redouble/pass
Table 5 provides a portion of Koca’s table, where it is player 1’s roll. The y-axis is
the peg(s) that player 1’s piece(s) are on and the x-axis is the peg(s) that player 2’s
piece(s) are on. Looking at the intersection between peg(s) on the y-axis and peg(s)
on the x-axis we get the doubling decision for player 1.
Table 5: Cube Actions for Money Play
1, 2, 3, 11, 12
4, 13
5
14
6
22
23
24, 15
25
1, 2, 3, 11, 12
5
5
5
5
5
4
4
4
4
4, 13
5
5
5
5
5
4
4
4
4
5
5
5
5
5
5
4
4
4
3
[5]
Example 15: Suppose that player 1 has two pieces on the 2-point and player 2 has
a piece on the 4-point. What cube decisions should each player make?
Using Table 5 we see that the intersection between player 1’s position and player 2’s
position is a 4. This cube decision is for player 1 to redouble and player 2 to accept
the double.
The cube decisions above are the ones we will compare to match play. For the entire
table of cube actions, refer to [5].
15
5.2
Match Play Doubling Decisions
Based on cube situations known for money play, we now look at the same situations
but in a match play situation. Notice that the differences in cube actions from money
play to match play are in bold in Table 6 and Table 7. First looking at Table 6, we
have the end game situation where player 1 is 2-away from winning the match and
player 2 is 3-away from winning the match.
Table 6: Cube Actions for Match Play 2-away, 3-away
1, 2, 3, 11, 12
4, 13
5
1, 4
6
2, 2
2, 3
24, 15
2, 5
1, 2, 3, 11, 12
5
5
5
5
5
5
4
4
4
4, 13
5
5
5
4
4
4
3
3
3
5
5
5
5
4
4
4
3
3
3
In Table 7 we look at the match play situation where player 1 is 2-away and player 2
is 4-away from winning the match.
Table 7: Cube Actions for Match Play 2-away, 4-away
1, 2, 3, 11, 12
4, 13
5
1, 4
6
2, 2
2, 3
24, 15
2, 5
1, 2, 3, 11, 12
5
5
5
5
5
5
4
4
4
4, 13
5
5
5
4
3
4
3
3
3
5
5
5
5
4
4
4
3
3
3
As we can see, the doubling decisions when player 2 changes from 3-away to 4-away
are very similar. From the three tables we can conclude that in money play the player
on roll can be more eager to double then in match play if they are ahead in the game.
In certain end game situations in match play the player has to wait a little longer
to double or sometimes has the chance to accept the double rather than decline it if
they are the opponent.
16
In order to complete the calculations for match play it is helpful to complete a tree
diagram for how the game can play out. Figure 2 shows the basic structure that the
tree diagram will have for calculating match equity of player 1. It can be used both
if player 1 doesn’t double and if player 1 doubles.
Figure 2
This tree diagram above can only be used in terms of player 1’s match equities where
probabilities for each situation occurring are placed on the stems on the tree. An
example of using the tree is below.
Example 16: Suppose player 1 has a piece on the 1-point and a piece on the 4-point
and player 2 has a piece on the 4-point. It is player 1’s turn to roll and there have
been no doubles thus far. Should player 1 double?
Solution: Figure 3 is a tree diagram for if player 1 doesn’t double. If player 1 does
not win, then we know that player 2 will double through a calculation involving the
equity for player 1 at that point in the tree. For this example that calculation will not
be included and it will be assumed that player 2 doubles at that point since player
2’s equity can be calculated to be higher if he doubles at this point as opposed to not
doubling.
17
Figure 3
Completing the match equity for player 1 if he doesn’t double can be done by using
the branches of the tree as follows:
7
7 2 34
29
(.75) + (.5) + ( )( )( ) = .71
36
36
36 36 36
A similar tree can be completed in the case that player 1 doubles. In this case, the
match equity for player 1 would be .87. Since the match equity is larger if player 1
doubles, this means player 1 should double.
Looking at player 2’s match equities for if he should accept or decline the double, we
see that if player 2 declines the double the match equity of player 2 is .25. This is
since the game would be over and the point would go to player 1, making the score
1-away, 3-away. Using the tree branch approach to calculate player 2’s match equity
if he accepts the double, we find that the match equity will be .32. This means that
player 2 should accept the double since his match equity would be higher in that
instance.
Looking at Table 6 we see that the 1-point, 4-point and 4-point position produces
the cube action double and accept, which is what our calculations showed. Table 5
shows us that in money play the decision will be that player 1 doubles and player 2
drops. Hence, one of the differences between the two types of game play.
18
6
Conclusions
In this paper, we have looked at when to double and when to accept a double in both
money play and match play Backgammon. In money play, it is practical to double
in a last move situation if a player’s probability of winning is greater than .50. In
match play, a player’s decision is based on how many points away each player is, and
can be calculated using match equities for each player. By comparing money play
and match play doubling situations, we have seen that in match play a player must
be more reluctant in their doubling decisions for 2-away, 3-away and 2-away, 4-away,
meaning they should not double as early as money play end situations.
The next step for this research project is to look at more end game situations involving
more complicated positions to see how doubling differs between money play and match
play. For example, it would be interesting to see whether the differences in 2-away,
3-away and 2-away, 4-away match play continue to 2-away, 5-away, etc. Further
research also includes if the same differences between money play and match play
occur if we are in 3-away, 2-away, 3-away, 3-away, etc.
7
Acknowledgements
I would like to thank my advisor Dr. Offner for all the time he has put into helping
me with my research. Also, thanks to my senior capstone class for all the suggestions
throughout the semester.
References
[1] Backgammon Equities. (2007). Gammoned. Retrieved from
http://www.gammoned.com/equities.html on 3 Sept. 2012.
[2] Backgammon History. (2007). Gammoned. Retrieved from
http://www.gammoned.com/history.html on 10 Sept. 2012.
[3] Gould, Ronald. (2010). Mathematics in Games, Sports, and Gambling. Boca
Raton, Florida: CRC Press. 3 Sept. 2012.
[4] Keeler, Emmet. (1975). Optimal Doubling in Backgammon. Backgammon
Galore. Retrieved from http://www.bkgm.com/articles/KeelerSpence/
OptimalDoublingInBackgammon/index.html on 13 Sept. 2012.
[5] Koca, Bob. (2007). Curing Your Short Bearoff Blues. Backgammon Galore.
Retrieved from http://www.bkgm.com/articles/Koca/CuringYourShortBearoff
Blues.html on 3 Sept. 13.
[6] Rules of Backgammon. (2012). Backgammon Galore. Retrieved from
http://www.bkgm.com/rules.html on 3 Sept. 2013.
19
[7] Woolsey, K. (1999). Five Point Match. GammOnLine. Retrieved from
http://www.bkgm.com/articles/GOL/Aug99/fivept.htm on 5 Nov. 2012.
20