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 Markers Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 1 Divider Chooser Method 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 Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 2 Divider Chooser Method 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 Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 3 Damian and Cleo Divide a Cheesecake 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 Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 4 Damian and Cleo Divide a Cheesecake 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 To ensure a fair division we assume neither of them knows anything about the other s likes and dislikes Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 5 Damian and Cleo Divide a Cheesecake 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 larger strawberry part Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 6 Damian and Cleo Divide a Cheesecake 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 twothirds 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 Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 7 Better to be the Chooser 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 Copyright 2010 Pearson Education Inc Excursions in Modern Mathematics 7e 3 2 8
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