Fair Division

1. Fair Division Divider-Chooser Method

2. Divider-Chooser T. Serino The Divider-Chooser method is, by far, the most popular fair division scheme. Without modification, this method only works for two players. Informally, it is referred to as the, you cut – I choose method. Given any continuous set S, the divider cuts S into two pieces, the chooser picks the piece he/she prefers, and the divider is left with the remaining piece.

3. Divider-Chooser T. Serino

4. Divider-Chooser T. Serino How does it work? Example: Well, let’s flip a coin to decide who divides and who chooses first. Let’s divide this candy bar. O.K. Call it in the air! Tails! O.K. You cut it, but I get to pick the first piece. Divider Chooser

5. Divider-Chooser T. Serino How does it work? Scenario 1: Hmmm, I think I should have done that differently. O.K. I’ve divided it, and labeled the pieces s1 and s2. Great! I pick s1. s1 s2

6. Divider-Chooser T. Serino How does it work? Scenario 2: O.K. I’ve divided it, and labeled the pieces s1 and s2. O.K. I pick s2. s1 s2 That’s fair!

7. Divider-Chooser T. Serino Scenario 1 didn’t seem fair at a glance, because the divider didn’t make a fair cut. (The divider was given the opportunity to make a fair cut.) Scenario 2 left both players with a fair share of the candy bar. The key to making this method work is that the divider must make a fair cut.

8. Divider-Chooser T. Serino When the divider makes a fair cut, it means that each share {s1, s2, s3, …sn} has exactly the same value as each of the other shares according to the divider’s value system. Scenario 2: Scenario 1: In this scenario, s1 and s2 were both valued the same by the divider. It was a fair cut. Unfair cut In this scenario, s1 was worth more to the divider than s2. It was not a fair cut. Fair cut

9. Divider-Chooser T. Serino Making fair cut can be a bit more challenging when different value systems are considered. Which of the following would be a fair cut for a divider who likes chocolate twice as much as he/she likes vanilla? S1 S2 S1 is only valued as 1/3 of the cake. S2 is valued as 2/3 of the cake. S1 S2 Not a fair cut.

10. Divider-Chooser T. Serino Making fair cut can be a bit more challenging when different value systems are considered. S1 Which of the following would be a fair cut for a divider who likes chocolate twice as much as he/she likes vanilla? S2 S1 is valued as exactly 1/2 of the cake. S2 is valued as exactly 1/2 of the cake. S1 S2 A fair cut.

11. Divider-Chooser T. Serino Making fair cut can be a bit more challenging when different value systems are considered. S2 Which of the following would be a fair cut for a divider who likes chocolate twice as much as he/she likes vanilla? S1 S1 is valued as exactly 1/2 of the cake. S2 is valued as exactly 1/2 of the cake. S1 S2 A fair cut.

12. Divider-Chooser T. Serino Example: Bianchy and Martin are going to divide a half chocolate, half vanilla cake using the divider-chooser method. The whole cake is worth $24. Bianchy likes vanilla cake twice as much as she likes chocolate and Martin likes chocolate three times as much as vanilla. Bianchy and Martin are new friends so neither of them has any knowledge of the other’s preference of flavor. $24

13. Divider-Chooser T. Serino Bianchy and Martin are going to divide a half chocolate, half vanilla cake using the divider-chooser method. Bianchy likes vanilla cake twice as much as she likes chocolate and Martin likes chocolate three times as much as vanilla. Bianchy and Martin are new friends so neither of them has any knowledge of the other’s preference of flavor. The couple decides to use the divider-chooser method to share the cake. Start by dividing each cake into what each player would consider fair slices. Bianchy likes vanilla cake twice as much as she likes chocolate Martin likes chocolate three times as much as vanilla Are these correct?

14. Divider-Chooser T. Serino $24 Determine the value of each share of this cake according to Bianchy’s value system. S1 S2

15. Divider-Chooser T. Serino $24 Determine the value of each share of this cake according to Martin’s value system. S2 S1

16. Divider-Chooser T. Serino If Bianchy were the divider, which of the following cuts would be fair and which would be unfair? s1 has 4 slices s2 has 2 slices s1 Unfair s1 has 3 slices s2 has 3 slices s1 Fair s1 has 3 slices s2 has 3 slices s1 Fair S1 S1 S1 S2 S2 S2

17. Divider-Chooser T. Serino If Martin were the divider, which of the following cuts would be fair and which would be unfair? s1 has 3 slices s2 has 5 slices s1 Unfair s1 has 4 slices s2 has 4 slices s1 Fair s1 has 4 slices s2 has 4 slices s1 Fair S1 S1 S1 S2 S2 S2

18. Try this. T. Serino Xavier likes strawberry cake twice as much as he likes chocolate. He likes vanilla cake three times as much as he likes chocolate. If Xavier were selected to be the divider, show two different ways that he can divide the 1/3 chocolate, 1/3 vanilla, 1/3 strawberry cake into two pieces that are both of equal value to him.

19. Try this. T. Serino Xavier likes strawberry cake twice as much as he likes chocolate. He likes vanilla cake three times as much as he likes chocolate. If Xavier were the chooser and the divider cut the cake as shown below, which piece would Xavier choose? Explain. S1 S2

20. Try this. T. Serino Xavier likes strawberry cake twice as much as he likes chocolate. He likes vanilla cake three times as much as he likes chocolate. c) If the cake is worth $36, what is the value of the share that Xavier chose in part b?

21. M athematical D ecision M aking