1. 3.1 Fair Division: To divide S into shares (one for each player) in such a way that each player gets a fair share. Fair Division: To divide S into shares (one for each player) in such a way that each player gets a fair share. Fair Share: Given a share s of S and a player P, we will say that s, in the opinion of P, is worth at least (1/N)th of the total value of S. (N) is the number of players. Fair Share: Given a share s of S and a player P, we will say that s, in the opinion of P, is worth at least (1/N)th of the total value of S. (N) is the number of players. To be continued… Discrete Math

2. 3.1 (Continued...) Continuous case: the set S is divisible in infinitely many ways. (Physically) Continuous case: the set S is divisible in infinitely many ways. (Physically) Discrete Case: Set S is made up of objects that are indivisible like houses/cars. Discrete Case: Set S is made up of objects that are indivisible like houses/cars. Discrete Math

3. 3.2 Divider Chooser Method: Made for two players, one player divides the object into two pieces, and the second player (chooser) picks the piece he/she wants, leaving the other piece to the divider. Divider Chooser Method: Made for two players, one player divides the object into two pieces, and the second player (chooser) picks the piece he/she wants, leaving the other piece to the divider. Discrete Math

4. 3.3 The Lone Divider Method: Steinhaus’ method for N=3 players. The Lone Divider Method: Steinhaus’ method for N=3 players. Step 1:Division: The divider (chosen at random) divides the cake into three pieces. Step 1:Division: The divider (chosen at random) divides the cake into three pieces. Step 2: Bidding: Player2 declares anonymously which of the three pieces are fair. Player3 does the same. Step 2: Bidding: Player2 declares anonymously which of the three pieces are fair. Player3 does the same. Step 3: Distribution: Step 3: Distribution: Case 1: Distribute each player a piece they think is fair if possible. They may swap at the end if it makes them happier. Case 1: Distribute each player a piece they think is fair if possible. They may swap at the end if it makes them happier. Case 2: If both P2 and P3 want the same piece, give P1 (divider) one of the unwanted pieces and then place the other two pieces back together in the divider-chooser method. Case 2: If both P2 and P3 want the same piece, give P1 (divider) one of the unwanted pieces and then place the other two pieces back together in the divider-chooser method. Discrete Math

5. 3.3 The Lone Divider for more than three players: The Lone Divider for more than three players: Case 1: Same Steps and distribute a fair share to each player. Case 1: Same Steps and distribute a fair share to each player. Case 2: A Standoff occurs: We first set aside the shares and players involved in the standoff. The remaining players can be assigned a fair share. The method is repeated for those players in the standoff by recombining the standoff shares. Case 2: A Standoff occurs: We first set aside the shares and players involved in the standoff. The remaining players can be assigned a fair share. The method is repeated for those players in the standoff by recombining the standoff shares. Discrete Math To be continued…

6. 3.4 The Lone Chooser Method (Three Players): One Chooser and Two Dividers: Step 1: Division: P1 and P2 divide S between themselves into two fair shares (divider-chooser method) Step 1: Division: P1 and P2 divide S between themselves into two fair shares (divider-chooser method) Step 2: Subdivision: P1 and P2 divide their shares into N subshares. Step 2: Subdivision: P1 and P2 divide their shares into N subshares. Step 3: Selection: The chooser [C] now selects one share from P1 and P2. These two subshares make up C’s final share. Step 3: Selection: The chooser [C] now selects one share from P1 and P2. These two subshares make up C’s final share. Discrete Math To be continued…

7. 3.4 (Continued...) The Lone Chooser for N players: The Lone Chooser for N players: Step 1: P 1 … P N-1 divide fairly the set S among themselves as if C does not exist. Step 1: P 1 … P N-1 divide fairly the set S among themselves as if C does not exist. Step2: Each divider subdivides his or her share into N subshares. Step2: Each divider subdivides his or her share into N subshares. Step 3: The chooser [C] finally gets to choose one subshare from each divider to get their fair share. Step 3: The chooser [C] finally gets to choose one subshare from each divider to get their fair share. Discrete Math

8. 3.5 The Last Diminisher (For any number of players) The Last Diminisher (For any number of players) Round 1: P1 starts by cutting what they believe to be exact (1/N) piece of S. P2 now has the opportunity to pass the piece or diminish it by trimming it to what they think it is (1/N)th of S. This process is continued by passing to each player. The player to last trim the piece will get the piece and be removed from the game. Round 1: P1 starts by cutting what they believe to be exact (1/N) piece of S. P2 now has the opportunity to pass the piece or diminish it by trimming it to what they think it is (1/N)th of S. This process is continued by passing to each player. The player to last trim the piece will get the piece and be removed from the game. Round 2: Repeat the process. If p1 is out (no one trimmed), then p2 resumes the cutting. Trimmings are added back to the remains. Round 2: Repeat the process. If p1 is out (no one trimmed), then p2 resumes the cutting. Trimmings are added back to the remains. The final two players will use the divider-chooser method. The final two players will use the divider-chooser method. Discrete Math

9. 3.6 Method of Sealed Bids: Estate division. Method of Sealed Bids: Estate division. Step 1: Bidding: Each player makes a secret bid on each item in the estate. Step 1: Bidding: Each player makes a secret bid on each item in the estate. Step 2: Allocation: Each item goes to the highest bidder. Step 2: Allocation: Each item goes to the highest bidder. Step 3: Payments: Calculate fair share (% of total bids). Subtract the money value of allocated items to get remaining claims. Step 3: Payments: Calculate fair share (% of total bids). Subtract the money value of allocated items to get remaining claims. Step 4: Dividing the Surplus: Add all remaining claims to acquire the money left over and divide this among players (%). Each player is then given their final settlement (list items won and money received or paid). Step 4: Dividing the Surplus: Add all remaining claims to acquire the money left over and divide this among players (%). Each player is then given their final settlement (list items won and money received or paid). Discrete Math To be continued…

10. 3.6 (Continued...) My Seven Steps: My Seven Steps: 1: Sum of Bids- add bids 1: Sum of Bids- add bids 2: Fair Share - % of them 2: Fair Share - % of them 3: Allocator - value of items awarded 3: Allocator - value of items awarded 4: Remaining Claims- (Fairshare - Allocations) 4: Remaining Claims- (Fairshare - Allocations) 5: Total Surplus: sum of all remaining claims (make positive) 5: Total Surplus: sum of all remaining claims (make positive) 6: Share of Surplus: % of total surplus. 6: Share of Surplus: % of total surplus. 7: Final Settlement: (Remaining claims and share of surplus) & List items Awarded. 7: Final Settlement: (Remaining claims and share of surplus) & List items Awarded. Discrete Math

11. 3.7 Method of Markers: Does not require players to put up any of their own money. Can’t be used effectively unless there are many more items to be divided than there are players, and items are close in value. Method of Markers: Does not require players to put up any of their own money. Can’t be used effectively unless there are many more items to be divided than there are players, and items are close in value. Step 1: Bidding: Line up items in an array. Each player will privately cut up the array string into N segments that they can see as an acceptable share. Step 1: Bidding: Line up items in an array. Each player will privately cut up the array string into N segments that they can see as an acceptable share. Discrete Math To be continued…

12. 3.7 (Continued...) Step 2: Allocation: Scanning the array from left to right, the first segment is given to the player whose marker is found first. These markers are now removed and the next segment goes to the player whose second marker is found first (only the segment between their two markers). This continues until each player has a segment. In case of a tie, break it randomly. Step 2: Allocation: Scanning the array from left to right, the first segment is given to the player whose marker is found first. These markers are now removed and the next segment goes to the player whose second marker is found first (only the segment between their two markers). This continues until each player has a segment. In case of a tie, break it randomly. Step 3: Dividing the leftovers: Randomly draw lots and let the players go in order picking one piece at a time or if enough pieces are left, do the method again. Step 3: Dividing the leftovers: Randomly draw lots and let the players go in order picking one piece at a time or if enough pieces are left, do the method again. Discrete Math