§ 3.7 The Method of Markers
Motivation
 We saw yesterday that the Method of Sealed
Bids worked well if all of the players had
enough money to play the game.
 In cases where this is not possible we will
need another method…
The Method of Markers
 The first thing we will do is arrange the goods
to be shared in a fixed ‘string’ called an array.
 Each player will independently bid for
segments of consecutive items in the array by
‘cutting’ the string.
 ‘Cuts’ will be made is by placing markers
along the array--each segment between a
player’s two markers will represent a fair
share of the loot to him/her.
Example: Martin (A), Bart (B), Millhouse
(C) andStep
Ralph
(D) stumble upon a box
1 (Bidding).
containing
a 15 pieces of old candy. They
Each player writes down independently on a
randomly piece
arrange
the candy
anhe/she
arraywants
and
of paper
exactly into
where
decide to his
useorthe
her method
markers. of markers to divide
their find.
In this case, there are 4 players so each player
needs to place 4 - 1 = 3 markers.
Example: Martin (A), Bart (B), Millhouse
(C) andStep
Ralph
(D) stumble upon a box
1 (Bidding).
containing
a 20 pieces of old candy. They
Each player writes down independently on a
randomly piece
arrange
the candy
anhe/she
arraywants
and
of paper
exactly into
where
decide to his
useorthe
her method
markers. of markers to divide
their find.
In this case, there are 4 players so each player
needs to place 4 - 1 = 3 markers.
The bids are opened and the results are shown
below:
D1 A1
A2 B1 A3 B2 B3 C1 D3 C2 C3
D2
Example: Martin (A), Bart (B), Millhouse
(C) andStep
Ralph
(D) stumble upon a box
2 (Allocations).
containing
a 20 pieces of old candy. They
Each player will be allocated one segment of
randomly the
arrange
into
array from
and
array. the
We candy
will begin
by an
scanning
decide to left
usetothe
method of markers to divide
right.
their find.
Here, the first marker from left-to-right is
Ralph`s (D1), so we give Ralph his first
segment. Now we can remove his
remaining markers.
D1 A1
A2 B1 A3 B2 B3 C1 D3 C2 C3
D2
Example: Martin (A), Bart (B), Millhouse
(C) andStep
Ralph
(D) stumble
upon a box
2 (Allocations).
Cont`d
containing
a 20 pieces of old candy. They
Now we look for the first second marker.
randomly arrange the candy into an array and
decide to use the method of markers to divide
In this case, Martin has the first second marker
their find.
(A2). We will give Martin his second
segment going from his first marker to the
second. We may then remove his
remaining markers.
A1
A2 B1 A3 B2 B3 C1
C2 C3
Example: Martin (A), Bart (B), Millhouse
(C) andStep
Ralph
(D) stumble
upon a box
2 (Allocations).
Cont`d
containing
a 20 pieces of old candy. They
Now we will look for the first third marker.
randomly arrange the candy into an array and
marker
(B3). We
give
decide This
to use
the belongs
methodtoofBart
markers
to will
divide
Bart his third segment going from his
their find. second marker to his third. We will also
now remove his markers.
B1
B2 B3 C1
C2 C3
Example: Martin (A), Bart (B), Millhouse
(C) andStep
Ralph
(D) stumble
upon a box
2 (Allocations).
Cont`d
containing
a 20 pieces of old candy. They
Finally, we will award Millhouse his final
randomly segment
arrange- the
candy after
into his
an third
arraymarker.
and
the portion
decide to use the method of markers to divide
their find.
C1
C2 C3
Example: Martin (A), Bart (B), Millhouse
(C) andStep
Ralph
(D) stumble
upon aCont`d
box
3 (Dividing
the Leftovers).
containing
a 20 pieces of old candy. They
Notice that there are still a few items leftover.
randomly These
arrange
candyby
into
an form
arrayofand
will the
be divided
some
decide to lottery.
use the method of markers to divide
their find.
The Method of Markers
(for N players and M objects)
 Step 1. Each player independently
divides the array into N fair shares by
placing N - 1 markers. The markers
separate each fair share from the next.
The Method of Markers
(for N players and M objects)
 Step 2. Scan the array from left-to-right
until the first first marker is located. The
player owning that marker gets to keep
his or her first segment and his/her
markers are removed. Continue moving
from left to right until the first second
marker is found. The player owning that
marker keeps his/her second segment.
Continue similarly until all players
receive one segment.
(Ties are broken randomly.)
The Method of Markers
(for N players and M objects)
 Step 3. The leftover items can be
divided among the players by some
form of lottery, or if there are many more
leftovers than players the method of
markers can be used again.