Fair Division
Lone Divider Method
Lone Divider method
The Lone Divider Method is a fair division
scheme that can be used to divide a bounty
between three or more players. As the name
implies, there is only one divider in the method
and the rest of the players are choosers (we’ll
call them chooser 1, chooser2, etc.).
Lone Divider method
Steps:
1. Decide which player is the divider. (roll a die, draw
straws, draw cards, buck up...)
2. The Divider cuts the cake into 3 slices (S1, S2, and S3).
The cut must be fair to the divider (Each slice must be
worth 1/3 of the whole cake).
3. Each of the choosers declares independently (secretly,
usually by writing it down on a slip of paper) which
slice(s) they each consider to be fair (worth at least 1/3)
4. The cake is distributed. (If the cake cannot be
distributed fairly at this point, other divisions may be
necessary)
Lone Divider method
Remember:
As in all fair division schemes, the Lone
Divider Method guarantees that each of
the n players (in this case n=3) will receive
what they consider a fair share.
A fair share is worth at least 1/n of the whole
cake (in this case, 1/3 of the
whole cake).
Lone Divider method
Example 1)
At least one of the choosers selects more than
one slice to be fair.
(possible to distribute the cake fairly without further division)
Scenario: The divider cuts the cake and labels
the slices as shown:
Chooser 1 declares: {S1, S3}
Chooser 2 declares: {S3}
Lone Divider method
Solution:
Chooser 1 declared: {S1, S3}
Chooser 2 declared: {S3}
The divider gets S2 (she declared this as a fair
slice by making the cut)
Chooser 2 gets S3
Chooser 1 gets S1
Lone Divider method
Example 2)
Each chooser selects one slice only, but they
each select a different slice.
(possible to distribute the cake fairly without further division)
Scenario: The divider cuts the cake and labels
the slices as shown:
Chooser 1 declares: {S1}
Chooser 2 declares: {S3}
Lone Divider method
Solution:
Chooser 1 declared: {S1}
Chooser 2 declared: {S3}
The divider gets S2 (she declared this as a fair
slice by making the cut)
Chooser 2 gets S3
Chooser 1 gets S1
Lone Divider method
Example 3)
Each chooser selects one slice only, and they
both select the same slice.
(Further division is necessary)
Scenario: The divider cuts the cake and labels
the slices as shown:
Chooser 1 declares: {S3}
Chooser 2 declares: {S3}
Lone Divider method
Solution:
Chooser 1 declared: {S3}
Chooser 2 declared: {S3}
The divider gets to choose either S1 or S2.
(Each chooser declared S1 and S2 as unfair.)
The remaining slices are put back together and
the choosers will use divider-chooser method to
divide it fairly.
Lone Divider method
*Important notes for understanding.
By selecting S3 as the only fair slice, the choosers are
implying that both S1 and S2 are each less than 1/3 of
the whole cake.
If, the slice taken by the divider is worth less than
1/3 of the cake, then the value of the remaining
slices is worth more than 2/3 of the cake.
Therefore, the choosers will each be guaranteed
more than their fair share of the cake.
Lone Divider method
Example 3)
(continued)
One possible scenario:
The Divider selects S1
S2 and S3 are combined.
The choosers use dividerchooser method.
Try this.
1 If you were the divider in a fair division
game with three players, show two
different ways that you can make a fair
division of the following rectangular cake.
Try this.
2 If you were one of the choosers, which
slices would you declare as fair in each of
the following examples?
S1
S1
S2
S2
S3
S3
Try this.
3 The divider has cut a cake as shown below.
Make a fair division of the cake given each
chooser’s declarations. (Decide who gets
which piece)
Chooser 1: {S1}
Chooser 2: {S1, S3}
S2
S1
S3
Lone Divider method