NAMES:
3.4 – The Method of Markers
When there are many objects to be divided among several players the Method of Markers may be
the best method of fair division. The procedure is as follows.
1. Placing the Markers. Arrange the objects in a row. Have each player independently place
his/her markers dividing the row into a number of equal-valued (according to the player)
portions. The number of portions should equal the number of players, hence there will be
one fewer marker than the number of players. {Four markers will divide the row into fifths,
etc.}
2. First Share. Give all objects up to the left-most marker to the player who placed that
marker. That player then has his/her share and is done. Remove all of his/her remaining
markers.
3. Second Share. Find the first occurring second marker. Suppose it was placed by player X.
Give X all objects between that marker and X’s first marker. X is now out, and X’s other
markers are removed.
4. Further Shares. Continue this process, looking for the first third marker, then the first
fourth, etc. When a single player is left, he/she gets the objects from his/her last marker to
the end of the row.
5. The Leftovers. Distribute the leftovers, if there are any, using a method of your choice.
1. (Warmup) Here are some objects (small letters to be exact) which have already been arranged in a
row. The four players A, B, C, & D have placed their markers. Draw boxes to indicate what the
markers placed by B indicate as a fair division into four portions.
a b
c | d
C
e
f | | g h | i
BC
A
j
k | l | m
D B
n | o
C
p| | q
AD
r | s
B
t
u | v
D
w | x
A
y z
2. Now follow the method of markers and draw boxes around the objects that go to each of the players
in the arrangement below, marking the boxes A, B, C, & D.
a b
c | d
C
e
f | | g h | i
BC
A
j
k | l | m
D B
n | o
C
p| | q
AD
There are _____ leftovers to be distributed as bonuses.
{More on other side}
r | s
B
t
u | v
D
w | x
A
y z
3. Four different players E, F, G, & H might divide up the same objects differently as shown. Once
again, draw boxes around the objects that go to each of the players and label them.
a b | c d | e
H
F
f | | g h
GE
i
j | k
F
l
m| n
G
o | p| |q
E EH
r | s
F
t | u
H
v | w
G
x
y z
There are _____ leftovers to be distributed as bonuses.
4. Find a different distribution scheme that could be applied to the objects in #3 using the same markers,
but still make everyone happy? As before, draw boxes around the objects that go to each of the
players.
a b | c d | e
H
F
f | | g h
GE
i
j | k
F
l
m| n
G
o | p| |q
E EH
There are _____ leftovers to be distributed as bonuses.
r | s
F
t | u
H
v | w
G
x
y z