§ 3.7 The Method of Markers

1
3.7 The Method of Markers
2
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

3
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 players two
  markers will represent a fair share of the loot
  to him/her.

4
Example Martin (A), Bart (B), Millhouse (C) and
Ralph (D) stumble upon a box containing a 15
pieces of old candy. They randomly arrange the
candy into an array and decide to use the method
of markers to divide their find.
Step 1 (Bidding). Each player writes down
independently on a piece of paper exactly where
he/she wants his or her markers. In this case,
there are 4 players so each player needs to place
4 - 1 3 markers.
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Example Martin (A), Bart (B), Millhouse (C) and
Ralph (D) stumble upon a box containing a 20
pieces of old candy. They randomly arrange the
candy into an array and decide to use the method
of markers to divide their find.
Step 1 (Bidding). Each player writes down
independently on a piece of paper exactly where
he/she wants his or her markers. 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
A1
A2
B1
D1
A3
B2
B3
D3
C1
C2
C3
D2
6
Example Martin (A), Bart (B), Millhouse (C) and
Ralph (D) stumble upon a box containing a 20
pieces of old candy. They randomly arrange the
candy into an array and decide to use the method
of markers to divide their find.
Step 2 (Allocations). Each player will be
allocated one segment of the array. We will
begin by scanning from left to right. Here, the
first marker from left-to-right is Ralphs (D1),
so we give Ralph his first segment. Now we can
remove his remaining markers.
A1
A2
B1
D1
A3
B2
B3
D3
C1
C2
C3
D2
7
Example Martin (A), Bart (B), Millhouse (C) and
Ralph (D) stumble upon a box containing a 20
pieces of old candy. They randomly arrange the
candy into an array and decide to use the method
of markers to divide their find.
Step 2 (Allocations). Contd Now we look for
the first second marker. In this case, Martin
has the first second marker (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
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Example Martin (A), Bart (B), Millhouse (C) and
Ralph (D) stumble upon a box containing a 20
pieces of old candy. They randomly arrange the
candy into an array and decide to use the method
of markers to divide their find.
Step 2 (Allocations). Contd Now we will look for
the first third marker. This marker belongs to
Bart (B3). We will give Bart his third segment
going from his second marker to his third. We
will also now remove his markers.
B1
B2
B3
C1
C2
C3
9
Example Martin (A), Bart (B), Millhouse (C) and
Ralph (D) stumble upon a box containing a 20
pieces of old candy. They randomly arrange the
candy into an array and decide to use the method
of markers to divide their find.
Step 2 (Allocations). Contd Finally, we will
award Millhouse his final segment - the portion
after his third marker.
C1
C2
C3
10
Example Martin (A), Bart (B), Millhouse (C) and
Ralph (D) stumble upon a box containing a 20
pieces of old candy. They randomly arrange the
candy into an array and decide to use the method
of markers to divide their find.
Step 3 (Dividing the Leftovers). Contd Notice
that there are still a few items leftover. These
will be divided by some form of lottery.
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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.

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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.)

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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.