Fair Division Worksheet Day 3. Warm-up 1. To cut a string into 2 pieces, how many times must it be cut?

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1 Name Fair Division Worksheet Day 3 Date Warm-up 1. To cut a string into 2 pieces, how many times must it be cut? 2. To cut a string into 3 pieces, how many times must it be cut? 3. To divide a string into n pieces, how many times must it be cut? 4. What is two times one third of one half? One person divides, the other chooses. 1. The lone divider divides the piece into three equal pieces. 2. The two choosers each bid on pieces that they think are at least 1/3 of the share. 3. Case 1: Each person is given a piece that he/she thinks is fair. Case 2: Each chooser bids on only the one same piece. i. The lone divider is given one of the pieces that both choosers thought was unfair. ii. The remaining pieces are recombined and divided by the choosers using divider-chooser. 5. C1 = { } C2 = { } Divider gets C1 gets C2 gets 6. Case 2: Bids: C1={s1} C2={s1}

2 7. Three players want to divide a cake fairly using the lone divider method. The divider cuts the cake into 3 slices. For each part, determine or describe the possible fair divisions of the cake. a. C1 = { s1, s2 } C2 = { s3 } D gets C1 gets C2 gets b. C1 = { s1, s2 } C2 = { s1, s2 } D gets C1 gets C2 gets c. C1 = { s3 } C2 = { s3 } 1. Two dividers divide using the divider chooser method. 2. Each divider divides his/her share into three equal parts. 3. The chooser picks one piece from each divider. 8. The dividers each get: Suppose the chooser thinks divider 1 s half is 48% and divider 2 s half is, then the chooser gets: Method of Markers for n items 1. The items to be divided are placed in a line left to right. The players are randomly given letters A, B, C... and ordered alphabetically. 2. Each person places n-1 markers to divide the line into n equal pieces. (These bids are done privately and then shared once all bids are made.) 3. i. To make the division, start at the left and move right to the first marker (if there is a tie go alphabetically). Give that player from the beginning of the line up to his/her first marker, and remove his/her remaining markers. ii. Now continue to move to the right and look for the first marker that has a 2 (if there is a tie go alphabetically). Give that player from his/her marker 1 to his/her marker 2 (and remove his/her remaining markers.) iii. Continue the process: one player gets from the beginning to marker 1, one player gets between marker 1 and 2, one player gets between marker 2 and 3,..., the last player gets from last marker to end. iv. The leftovers are giving out in a lottery of the players.

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4 The players are randomly placed in a line. The first person in line cuts what he/she thinks is a fair piece. Then each person in the order of the line gets a chance to either say it is fair, or cut it to make it fair. If a person cuts the piece, it is now his/her piece. The last person to cut (diminish) the piece gets the piece. The process continues n-1 times until everyone has a piece. There are n -1 rounds because the last person will get what remains people (P1, P2, P8) are dividing a cake using the Last Diminisher Method. Each round has the following diminishers: R1: P4, P5, P6 R2: No-one R3: P3, P8 R4: Anyone that can diminish does. R5: P4 a. Who cuts the piece in round 1? b. Who gets the piece in round 1? c. Who cuts the piece in round 2? d. Who gets the piece in round 2? e. Who cuts the piece in round 3? f. Who gets the piece in round 3? g. Who is the last person with the opportunity to diminish the piece in round 4? i. Who gets the piece in round 4? j. Who cuts the piece in round 5? k. Who gets the piece in round 5? l. Who cuts the piece in round 6? m. How many rounds does it take to divid the cake among 8 people?

5 14. Five players want to divide a cake fairly using the lone divider method. The divider cuts the cake into 5 slices. For each part, determine or describe the possible fair divisions of the cake. a. C1 = { s1, s2 } C2 = { s3 } C3 = { s4 } C4 = { s1} b. C1 = { s2 } C2 = { s1, s3 } C3 = { s3, s4 } C4 = { s1, s5 }

6 people (P1, P2, P6) are dividing a cake using the Last Diminisher Method. Each round has the following diminishers: R1: P4, P5, P6 R2: No-one R3: P3 R4: Anyone that can diminish does. a. Who cuts the piece in round 1? b. Who gets the piece in round 1? c. Who cuts the piece in round 2? d. Who gets the piece in round 2? e. Who cuts the piece in round 3? f. Who gets the piece in round 3? g. Who is the last person with the opportunity to diminish the piece in round 4? h. Who gets the piece in round 4? i. Who cuts the piece in round 5? j. How many rounds does it take to divid the cake among 6 people?