Transcript Ch 3 powerpoint

Fair Division
•
Fair Division Problem: A problem that
involves the dividing up of an object or set of
objects among several individuals (players) so
that each individual considers the part he or
she receives to be a fair portion.
• Assumptions:
– Cooperation: players are willing
participants
– Rationality: players are rational
– Privacy: players have no info on other
players
– Symmetry : players have equal rights
copyright 2005
Dr. Annette M. Burden
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Fair Division - Objectives
1. Fair Share: each of the N individuals
gets what he/she considers a fair 1/N
portion of the whole.
1. What may be considered a fair share by 1
player may not be considered a fair share
by another player
2. It is possible for a player to get a fair share
portion but not the preferred piece.
2. Envy Free: Each player gets a piece of
the whole that he/she considers is the
best share
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Dr. Annette M. Burden
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Fair Division - Types
Origins go back 5,000 years. Modern era of
fair division in math began in Poland
during WWII
1. Continuous: The item(s) can be divided
many ways & by small amounts (pie,
cake, land, etc.)
2. Discrete: The item(s) consist of objects
that cannot be split up (boat, book, etc.)
3. Mixed: The item(s) are both continuous
and discrete
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Dr. Annette M. Burden
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Continuous Methods - Overview
1. Divider - Chooser: 2 player game.
One player cuts, other chooses.
2. Lone Divider: 3 player game. One
player cuts, two players choose.
3. Lone Chooser 3 player game. Two
players cut, one player chooses.
4. Last Diminisher: More than 3
players
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Dr. Annette M. Burden
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Basic Concept
Chocolate
Strawberry
400
200
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Jerry buys a chocolate-strawberry cake for $20.
Jerry values chocolate 4 times as much as
he values strawberry.
1. What is the value of the strawberry ½ of
the cake?
a) 4x + x = $20 or 5x = 20 or x = $4
2. What is the value of the chocolate ½ of the
cake?
a) 4($4) = $16
3. A piece of the cake is cut as shown at left.
What is the value of the piece to Jerry?
4. A piece of the cake is cut as shown at left.
What is the value of the piece to Jerry?
a)
($16)(40/180) + ($4)(20/180) =
$4.00
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Basic Concept
Whole Cake
s1
s2
s3
Andy
$12.00
$3.00
$5.00
$4.00
Paul
$15.00
$4.00
$4.50
$6.50
Cheryl
$13.50
$4.50
$4.50
$4.50
Which of the three slices are fair shares to:
•Andy: $12/3 players = $4 so any piece over $4 or s2,s3
•Paul: $15/3 players = $5 so any piece over $5 or s3
•Cheryl:$13.50/3 players = $4.50 so any piece over $4.50 or s1,s2,s3
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Dr. Annette M. Burden
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Divider – Chooser Method
i.
Player A divides cake into 2 parts in any way
he or she desires.
ii.
Player B chooses the piece he or she wants.
iii. Envy-free scheme.
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Dr. Annette M. Burden
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Steinhaus Lone-Divider Procedure: 3 players
i.
By random draw, one of the 3 players is designated
to be the divider, D. The other 2 players will be
choosers, C1 and C2
D divides the cake into 3 pieces s1,s2,s3 as equal as
possible.
C1 declares which of the 3 pieces are a fair share to
him. Independently, C2 does the same. These are the
choosers’ bids.
How the pieces are divided:
ii.
iii.
iv.
i.
ii.
Case I: if c1 and c2 like different pieces then D gets the piece
neither of the choosers want
Case II: if c1 and c2 both like the same piece and both
disapprove of the same piece, then D gets the piece that the
choosers both disapprove of. The remaining pieces are put
together and the divide and choose method is used to
determine who gets what.
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Dr. Annette M. Burden
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Steinhaus Lone-Divider Procedure: 3 players
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Dr. Annette M. Burden
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Steinhaus Lone-Divider Procedure: 3 players
s1
s2
s3
D
33 1/3 %
33 1/3 %
33 1/3 %
c1
35%
10%
55%
c2
40%
25%
35%
s1
s2
s3
D
33 1/3 %
33 1/3 %
33 1/3 %
c1
30%
40%
30%
c2
60%
15%
25%
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Dr. Annette M. Burden
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Lone Chooser Procedure
3 players
i.
Players A and B divide the cake using
the Divide and Choose method
ii. Player A then subdivides her share into
thirds while player B subdivides his
share into thirds.
iii. Player C then selects a 1/3rd share from
player A and a 1/3rd share from player
B.
iv. Players A & B keep their remaining
shares.
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Dr. Annette M. Burden
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Lone Chooser
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Last Diminisher (4 player example)
• A cuts ¼ of cake and hands it to B
• If B feels piece > ¼, B trims piece placing trimmings with
remainder of cake and passes cut piece to C. If B feels piece
is < to ¼, then B passes piece to C without trimming it.
• With the piece given to him by B, C does same procedure as
B, passing the trimmed or untrimmed piece onto D.
• D does same as other players, but D is the last player, so if D
trims the piece, D keeps it and exits the game. Otherwise, the
piece goes back to the last player who trimmed it or to A if
no one trimmed it and that player exits the game.
• Process starts over by cutting another piece of
approximately 1/3 of the original from the remaining cake.
• When down to 2 players, use divide and choose scheme.
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Dr. Annette M. Burden
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Last Diminisher
• Ann – cuts ¼ of cake
• Bob- trims the piece
• Carl - pass
• Deb- pass
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Bob claims the slice and is out
Of the game.
Dr. Annette M. Burden
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Last Diminisher
• Ann – cuts ¼ of cake
• Carl - pass
• Deb- pass
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Ann gets the slice of cake
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Last Diminsher
• Randomly select a divider between Carl and
Deb.
• Use divider-chooser method to finish
dividing the cake
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Dr. Annette M. Burden
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Discrete Methods - Overview
1. Method of sealed bids: Used
primarily to divide up an
inheritance
2. Method of Markers: Used
primarily to divide up many
items that are similar in value
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Dr. Annette M. Burden
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Method of Sealed Bids
1. Each player independently assigns a value all
assets to be divided.
2. Award items to the highest bidders.
3. Compute each person’s total by adding each
person’s bids of the items.
4. Compute each person’s fair share by dividing
his/her total by the total number of bidders.
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Method of Sealed Bids
5. Subtotal for each bidder =
Item awarded – (player’s value of item awarded – fair share)
6. Surplus =
fair share – sum of each player’s value of item awarded
7. Extra = surplus divided by total number of bidders.
8. Final Division for each bidder =
Item awarded – (player’s value of item awarded – fair share) +
extra
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Dr. Annette M. Burden
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Method of Sealed Bids
Object
James
Art
Martha
Cleo
House
$80.000
$75,000
$90,000
$60,000
Car
$10,000
$12,000
$13,000
$15,000
Totals
$90,000
$87,000
$103,000
$75,000
Fair Share
$22,500
$21,750
$25,750
$18,750
$22,500
$21,750
House – (90,000 - Car –($15,000 25,750)
18,750)
(Totals/4)
Subtotal
Surplus in estate: $64,250 - $3,750 - $22,500 - $21,750 = $16,250
Extra awarded to each: $16,250/4 = $4,062
Totals
$22,500
$21,750
House - $64,250 Car + $3,750
surplus/4
$4,062.50
$4,062.50
$4,062.50
$4,062.50
Final
Division
$26,565.50
$25,812.50
House $60,187.50
Car -$7,812.50
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Method of Markers
1. The items are laid out in a row with no
particular ordering of the items in mind.
2. (Bidding) Each player independently divides
the row into N fair shares by placing N-1
markers between the items.
3. (Allocations) Scan the array from left to right
until the first 1st marker of any player is
located. That player is allocated the items up
to his/her 1st marker and the player exits.
His/her remaining markers are removed.
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Dr. Annette M. Burden
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Method of Markers
4. Continue the Allocation procedure scanning
from left to right until the first 2nd marker of
any player is located. That player is allocated
the items from his/her 1st marker to his/her 2nd
marker. His/her remaining markers are
removed and that player exits. Continue in
this manner until all players have been
allocated a fair portion of the items.
5. (Leftovers) The leftover items can be divided
among the players via a lottery procedure.
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Method of Markers
A3
A2
A1
C1
D1
B1
C2
B2
D2
B3
D3
C3
4 players so there should be 3 divisions markers per player
Black arrows: player A. Red arrows: player B. Blue arrows: player C.
Green arrows: Player D
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Dr. Annette M. Burden
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Method of Markers
A3
A2
A1
C1
D1
B1
C2
B2
D2
B3
D3
C3
Player A (Black) is the first marker encountered, so player A gets the
Grouping taking him to his 1st marker. Player A exits the game and
The remainder of the black markers are removed.
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Dr. Annette M. Burden
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Method of Markers
C1
D1 B1
C2
B2
D2
B3
D3
C3
Player C (Blue) is the first 2nd marker encountered, so player C gets
the items between the 1st blue and 2nd blue markers. Player C exits the
Game and the remainder of the blue markers are removed.
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Dr. Annette M. Burden
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Method of Markers
D1
B1
B2
D2
B3 D3
Player B (Red) is the first 3rd marker encountered, so player B gets
the items between the 2nd red and 3rd red markers. Player B exits the
Game and the remainder of the red markers are removed.
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Dr. Annette M. Burden
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Method of Markers
D1
D2
D3
Player D (Green) gets the last group of items from the green 3rd marker
To the end of the row. The remaining 2 items are divided evenly
Among the 4 players.
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Credits
• Tannenbaum, Excursions in Modern
Mathematics, 5th ed
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Dr. Annette M. Burden
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