Lone-Divider Method
Day 2 – Fair Division
Requirements
3
or more players
 Continuous set of goods to be
divided
Procedure with 3 players
1.
Randomly select one divider. The others will
act as choosers (C1 and C2)
(For example….rolling a dice, drawing a card, etc.)
2.
The divider will divide the booty into what
he/she believes to be 3 fair pieces.
3.
C1 and C2 bid on which pieces they believe
to be fair (worth at least 1/3 of total)
(Bids are done independently. Write on separate sheets of
paper and then compare)
Procedure with 3 players, cont’d
4. a) If C1 and C2 have both bid on two
different pieces, C1 and C2 will get the
pieces they voted for and the divider will
get the remaining piece.
b) If C1 and C2 have bid on the same
piece, then the divider is given a piece
that neither C1 and C2 want and the two
remaining pieces are then combined and
the divider-chooser method is used.
Example 1
Dale, Cindy, and Cher are
dividing a cake. Dale is
the divider and divides
the cake into three
pieces s1, s2, and s3.
s1
s2
s3
Dale 33 13 % 33 13 % 33 13 %
Cindy 35% 10% 55%
Cher 40%
Which pieces will Cindy bid for?
s3, s1
Which pieces will Cher bid for?
s1, s3
Which piece will Dale get?
s2
Which piece will Cher and Cindy get?
Cher - s1
Cindy - s3
25% 35%
Example 2
Dale, Cindy, and Cher are
dividing a cake. Dale is
the divider and divides
the cake into three
pieces s1, s2, and s3.
s1
s2
s3
Dale 33 13 % 33 13 % 33 13 %
Cindy 30% 40% 30%
Cher 60% 15%
Which pieces will Cindy bid for?
s2
Which pieces will Cher bid for?
s1
Which piece will Dale get?
s3
Which piece will Cher and Cindy get?
Cher – s2
Cindy - s1
25%
Fair Division


The Lone-Divider Method for Three
Players
Step 3 ( Distribution). Who gets the
piece? The answer depends on the bid
lists. For convenience, we will separate
the pieces into two groups: chosen
pieces (let’s call them C- pieces), and
unwanted pieces (let’s call them Upieces).
7
Fair Division
The Lone-Divider Method for Three Players
The chart shows how Dale divided the cake, and how
Cindy and Cher value the 3 pieces.
s1
s2
s3
Dale
33⅓%
33⅓%
33⅓%
Cindy
20%
30%
50%
Cher
10%
20%
70%
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Example 3
s1
s2
s3
Dale
33⅓%
33⅓%
33⅓%
Cindy
20%
30%
50%
Cher
10%
20%
70%
Since Cindy and Cher value s1 the
least, they give it to Dale. They then
take the remaining pieces and divide
them using the Divider-Chooser
method.
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Example 4 w/ more than 3
players
Demi
Chan
Chloe
Chris
s1
25%
30%
20%
25%
s2
25%
20%
20%
20%
s3
25%
35%
40%
20%
s4
25%
15%
20%
35%
Who is the divider, and who are the choosers?
Demi is our divider because she is the only one
who has equal distribution of the pieces. Chan,
Chloe, and Chris will be the choosers.
Example 4 w/ more than 3
players
s
s
What is Chan’s bid?
s1, s3
What is Chloe’s bid?
s3
What is Chris’s bid?
s1, s4
Which piece does that leave
for Demi?
s2
1
2
s3
s4
Demi
25%
25%
25%
25%
Chan
30%
20%
35%
15%
Chloe
20%
20%
40%
20%
Chris
25%
20%
20%
35%
Chloe will have to get s3
Which means Chan will have to get s1
And Chris is left with s4
Example 5
We have a half chocolate and half strawberry cake worth $24.
The Divider likes chocolate twice as much as strawberry, but
both Choosers like chocolate and strawberry equally. Let’s
say the Divider cuts the cake like the picture below. Now, the
choosers value s1 to be $12 and each of s2 and s3 to be
worth $6. Only s1 is fair to both chooser.
So what happens now?
Divider takes one of the chocolate pieces since that is of lesser value to the chooser and
the other 2 pieces are put together for Divider-Chooser method.
How much is the remaining cake worth to each of the Choosers?
All strawberry and half chocolate is worth $18 to the choosers
When the choosers play Divider-Chooser, what is the minimum value of the slices that the Choosers
will end up with?
Half of $18 is $9
Ok. So the Divider is out of the game. (He’s got his piece) Let’s say that Chooser I is designated the
Divider, and Chooser II is the Chooser (because they’re playing Divider-Chooser now). What’s the
simplest correct cut Chooser I can make?
What are the values of the slices the two Choosers will get in this scenario?
Strawberry piece
Half Chocolate and a little Strawberry piece
Lone-Chooser Method
Requirements
3
or more players
 Continuous set of goods to
divide
Procedure with 3 players
1.
Randomly select one chooser and two dividers (D1
and D2)
2.
D1 and D2 divide the goods into s1 and s2 using the
Divider-Chooser method.
3.
D1 and D2 then divide their fair share into what they
believe to be 3 equal shares each. Call these s1a, s1b,
s1c, s2a, s2b, s2c.
4.
The chooser then selects one of {s1a, s1b, s1c} and
one of {s2a, s2b, s2c}. D1 and D2 will keep their
remaining shares.
Fair Division
The Lone-Chooser Method for Three
Players
 Step 1 ( Division). D1 and D2 divide S
between themselves into two fair shares.
To do this, they use the divider-chooser
method. Let’s say that D1 ends with S1
and D2 ends with S2 .

18
Fair Division


The Lone-Chooser Method for Three
Players
Step 2 (Subdivision). Each divider
divides his or her share into three
subshares. Thus D1 divides S1 into three
subshares, which we will call S1a, S1b and
S1c . Likewise, D2 divides S2 into three
subshares, which we will call S2a,
S2b and S2c .
19
Fair Division


The Lone-Chooser Method for Three
Players
Step 3 (Selection). The chooser C
now selects one of D1 ‘s three
subshares and one of D2 ‘s three
subshares. These two subshares
make up C’s final share. D1 then
keeps the remaining two subshares
from S1 , and D2 keeps the remaining
two subshares from S2 .
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Example 1
Let’s take our $24 cake that’s half strawberry and half chocolate. Divider 1 likes
strawberry and chocolate equally. Divider 2 likes chocolate twice as much as
strawberry. The Chooser likes strawberry twice as much as chocolate. Divider 1
cuts the cake like the picture below. Since these slices have the same value to
Divider 2, it doesn’t matter which slice Divider 2 picks. Let’s suppose he picks
the right slice.
Find simple, legal cuts for the two dividers so that each has equal
value to each divider.
Example continued.
Which 2 pieces would the Chooser pick?
Find the values for the pieces that the players get.