3.1 fair division: to divide s into shares (one for each player) in such a way that each player gets...

3.13.1 Fair Division: To divide S into shares (one Fair Division: To divide S into shares (one

for each player) in such a way that each for each player) in such a way that each player gets a fair share.player gets a fair share.

Fair Share: Given a share s of S and a Fair Share: Given a share s of S and a player P, we will say that s, in the opinion player P, we will say that s, in the opinion of P, is worth at least (1/N)th of the total of P, is worth at least (1/N)th of the total value of S. (N) is the number of players.value of S. (N) is the number of players.

To be continued…

Discrete MathDiscrete Math

3.1 (Continued...)3.1 (Continued...) Continuous case: the set S is divisible in Continuous case: the set S is divisible in

infinitely many ways. (Physically)infinitely many ways. (Physically) Discrete Case: Set S is made up of objects Discrete Case: Set S is made up of objects

that are indivisible like houses/cars.that are indivisible like houses/cars.


3.23.2 Divider Chooser Method: Made for two Divider Chooser Method: Made for two

players, one player divides the object into players, one player divides the object into two pieces, and the second player two pieces, and the second player (chooser) picks the piece he/she wants, (chooser) picks the piece he/she wants, leaving the other piece to the divider.leaving the other piece to the divider.


3.33.3 The Lone Divider Method: Steinhaus’ method The Lone Divider Method: Steinhaus’ method

for N=3 players.for N=3 players.Step 1:Division: The divider (chosen at random) Step 1:Division: The divider (chosen at random)

divides the cake into three pieces.divides the cake into three pieces.Step 2: Bidding: Player2 declares anonymously Step 2: Bidding: Player2 declares anonymously

which of the three pieces are fair. Player3 does the which of the three pieces are fair. Player3 does the same.same.

Step 3: Distribution:Step 3: Distribution: Case 1: Distribute each player a piece they think is Case 1: Distribute each player a piece they think is

fair if possible. They may swap at the end if it makes fair if possible. They may swap at the end if it makes them happier.them happier.

Case 2: If both P2 and P3 want the same piece, give Case 2: If both P2 and P3 want the same piece, give P1 (divider) one of the unwanted pieces and then P1 (divider) one of the unwanted pieces and then place the other two pieces back together in the place the other two pieces back together in the divider-chooser method.divider-chooser method.


3.33.3 The Lone Divider for more than three The Lone Divider for more than three

players:players:Case 1: Same Steps and distribute a fair share Case 1: Same Steps and distribute a fair share

to each player.to each player.Case 2: A Standoff occurs: We first set aside Case 2: A Standoff occurs: We first set aside

the shares and players involved in the standoff. the shares and players involved in the standoff. The remaining players can be assigned a fair The remaining players can be assigned a fair share. The method is repeated for those share. The method is repeated for those players in the standoff by recombining the players in the standoff by recombining the standoff shares.standoff shares.


To be continued…

3.4 3.4 The Lone Chooser Method (Three The Lone Chooser Method (Three Players): One Chooser and Two Players): One Chooser and Two Dividers:Dividers:

Step 1: Division: P1 and P2 divide S between Step 1: Division: P1 and P2 divide S between themselves into two fair shares (divider-themselves into two fair shares (divider-chooser method)chooser method)

Step 2: Subdivision: P1 and P2 divide their Step 2: Subdivision: P1 and P2 divide their shares into N subshares.shares into N subshares.

Step 3: Selection: The chooser [C] now selects Step 3: Selection: The chooser [C] now selects one share from P1 and P2. These two one share from P1 and P2. These two subshares make up C’s final share.subshares make up C’s final share.


To be continued…

3.4 (Continued...)3.4 (Continued...) The Lone Chooser for N players:The Lone Chooser for N players:

Step 1: PStep 1: P11… P… PN-1N-1 divide fairly the set S among divide fairly the set S among themselves as if C does not exist.themselves as if C does not exist.

Step2: Each divider subdivides his or her share Step2: Each divider subdivides his or her share into N subshares.into N subshares.

Step 3: The chooser [C] finally gets to choose Step 3: The chooser [C] finally gets to choose one subshare from each divider to get their fair one subshare from each divider to get their fair share.share.


3.53.5 The Last Diminisher (For any number of The Last Diminisher (For any number of

players)players)Round 1: P1 starts by cutting what they believe Round 1: P1 starts by cutting what they believe

to be exact (1/N) piece of S. P2 now has the to be exact (1/N) piece of S. P2 now has the opportunity to pass the piece or diminish it by opportunity to pass the piece or diminish it by trimming it to what they think it is (1/N)th of S. trimming it to what they think it is (1/N)th of S. This process is continued by passing to each This process is continued by passing to each player. The player to last trim the piece will get player. The player to last trim the piece will get the piece and be removed from the game.the piece and be removed from the game.

Round 2: Repeat the process. If p1 is out (no one Round 2: Repeat the process. If p1 is out (no one trimmed), then p2 resumes the cutting. trimmed), then p2 resumes the cutting. Trimmings are added back to the remains.Trimmings are added back to the remains.

The final two players will use the divider-chooser The final two players will use the divider-chooser method.method.


3.63.6 Method of Sealed Bids: Estate division.Method of Sealed Bids: Estate division.

Step 1: Bidding: Each player makes a secret bid Step 1: Bidding: Each player makes a secret bid on each item in the estate.on each item in the estate.

Step 2: Allocation: Each item goes to the Step 2: Allocation: Each item goes to the highest bidder.highest bidder.

Step 3: Payments: Calculate fair share (% of Step 3: Payments: Calculate fair share (% of total bids). Subtract the money value of total bids). Subtract the money value of allocated items to get remaining claims.allocated items to get remaining claims.

Step 4: Dividing the Surplus: Add all remaining Step 4: Dividing the Surplus: Add all remaining claims to acquire the money left over and claims to acquire the money left over and divide this among players (%). Each player is divide this among players (%). Each player is then given their final settlement (list items won then given their final settlement (list items won and money received or paid).and money received or paid).


To be continued…

3.6 (Continued...)3.6 (Continued...) My Seven Steps:My Seven Steps:

1: Sum of Bids- add bids1: Sum of Bids- add bids2: Fair Share - % of them2: Fair Share - % of them3: Allocator - value of items awarded3: Allocator - value of items awarded4: Remaining Claims- (Fairshare - Allocations)4: Remaining Claims- (Fairshare - Allocations)5: Total Surplus: sum of all remaining claims 5: Total Surplus: sum of all remaining claims

(make positive)(make positive)6: Share of Surplus: % of total surplus.6: Share of Surplus: % of total surplus.7: Final Settlement: (Remaining claims and 7: Final Settlement: (Remaining claims and

share of surplus) & List items Awarded.share of surplus) & List items Awarded.


3.73.7 Method of Markers: Does not require Method of Markers: Does not require

players to put up any of their own money. players to put up any of their own money. Can’t be used effectively unless there are Can’t be used effectively unless there are many more items to be divided than there many more items to be divided than there are players, and items are close in value.are players, and items are close in value.

Step 1: Bidding: Line up items in an array. Each Step 1: Bidding: Line up items in an array. Each player will privately cut up the array string into player will privately cut up the array string into N segments that they can see as an acceptable N segments that they can see as an acceptable share.share.


To be continued…

3.7 (Continued...)3.7 (Continued...)Step 2: Allocation: Scanning the array from left Step 2: Allocation: Scanning the array from left

to right, the first segment is given to the player to right, the first segment is given to the player whose marker is found first. These markers are whose marker is found first. These markers are now removed and the next segment goes to now removed and the next segment goes to the player whose second marker is found first the player whose second marker is found first (only the segment between their two markers). (only the segment between their two markers). This continues until each player has a segment. This continues until each player has a segment. In case of a tie, break it randomly.In case of a tie, break it randomly.

Step 3: Dividing the leftovers: Randomly draw Step 3: Dividing the leftovers: Randomly draw lots and let the players go in order picking one lots and let the players go in order picking one piece at a time or if enough pieces are left, do piece at a time or if enough pieces are left, do the method again.the method again.


3.1 fair division: to divide s into shares (one for each player) in such a way that each player gets...

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