Excursions in Modern Mathematics, 7e: 3.2 - 1 Copyright © 2010 Pearson Education, Inc. 3 The Mathematics of Sharing 3.1 Fair-Division Games 3.2 Two Players: The Divider-Chooser Method 3.3 The Lone-Divider Method 3.4 The Lone-Chooser Method 3.5 The Last-Diminsher Method 3.6 The Method of Sealed Bids 3.7 The Method of MarkersExcursions in Modern Mathematics, 7e: 3.2 - 2 Copyright © 2010 Pearson Education, Inc. The divider-chooser method (also called the you cut–I choose method) can be used when the fair-division game involves two players and a continuous set S As this name suggests, one player, called the divider, divides S into two shares, and the second player, called the chooser, picks the share he or she wants, leaving the other share to the divider.!Divider-Chooser MethodExcursions in Modern Mathematics, 7e: 3.2 - 3 Copyright © 2010 Pearson Education, Inc. This method guarantees that divider and chooser will each get a fair share (with two players, this means a share worth 50% or more of the total value of S). !Not knowing the chooser’s likes and dislikes (privacy assumption), the divider can only guarantee himself a 50% share by dividing S into two halves of equal value (rationality assumption); the chooser is guaranteed a 50% or better share by choosing the piece he or she likes best. !Divider-Chooser MethodExcursions in Modern Mathematics, 7e: 3.2 - 4 Copyright © 2010 Pearson Education, Inc. On their first date, Damian and Cleo go to the county fair. They buy jointly a raffle ticket and win a half chocolate–half strawberry cheesecake. Damian likes chocolate and strawberry equally well, so in his eyes the chocolate and strawberry halves are equal in value. Damian and Cleo Divide a CheesecakeExcursions in Modern Mathematics, 7e: 3.2 - 5 Copyright © 2010 Pearson Education, Inc. However, Cleo hates chocolate so the chocolate part of the cake is worth 0% of the whole cake, and the strawberry part is worth 100% of the whole cake (as far as Cleo is concerned.!Damian and Cleo Divide a Cheesecake To ensure a fair division, we assume neither of them knows anything about the other’s likes and dislikes.Excursions in Modern Mathematics, 7e: 3.2 - 6 Copyright © 2010 Pearson Education, Inc. Damian volunteers to go first (the divider). According to Damien’s value system, any physical half of the cake is a fair share, so he cuts the cake into two halves, ignoring the amount of strawberry / chocolate in either half. It is now Cleo’s turn to choose, and her choice is obvious: she will pick the piece having the Damian and Cleo Divide a Cheesecake larger strawberry part.Excursions in Modern Mathematics, 7e: 3.2 - 7 Copyright © 2010 Pearson Education, Inc. Final outcome: Damian gets a piece that is worth exactly half of the cake (According to Damian’s value system) Cleo ends up with a much sweeter deal–a piece that in her own eyes is worth about two-thirds of the cake. (According to Cleo’s value system) This is a fair division of the cake–both players get pieces worth 50% or more (according to their respective value systems)!Damian and Cleo Divide a CheesecakeExcursions in Modern Mathematics, 7e: 3.2 - 8 Copyright © 2010 Pearson Education, Inc. This example illustrates that it is better to be the chooser than the divider. The divider is guaranteed a share worth exactly 50% of the total value of S, The chooser could end up with a share worth more than 50%. (If the players each had the same value system they would each end up with exactly 50%. The differences between their value systems is what allows the chooser to (potentially) end up with more than 50%) Better to be the
View Full Document
Unlocking...