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class+February+22chapter3sharing

class+February+22chapter3sharing - Math 103 Section 11...

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Math 103, Section 11, Tuesday, February 22, 2011 Chapter 3 Handout 1 The Mathematics of Sharing Assignment Chapter 3A Please do Exercises 12,18,30, and 32. Homework will be submitted to Sakai. Homework to be discussed in class on Friday February 25. Exercise 7,9, and 11 This is an adorable divider-chooser video on youtube! Please watch it. http://www.youtube.com/watch?v=AdYFVN35h5w . Chapter 3 3.1 Fair-Division Games: The Mathematics of Sharing Basic elements of a fair-division game: 1. The goods or booty Examples: cake, pizza, jewelry, art, land, water rights, drilling rights, chores 2. The players Examples: individuals, clubs, political parties, states, nations 3 The value systems Examples: Each player can look at the set S or any part of the set S and assign to it a value. Either in absolute terms (to me, that’s worth $3.00) or in relative terms (to me, that piece is worth 25% of the total value of S). Basic Assumptions: 1. Rationality-each player wants to maximize his or her share of the booty S. Player uses reason and no psychology, mind games, etc. 2. Cooperation. Players accept the rules of the game. No judges or referees. 3. Privacy. Players have no useful information on the other players’ value systems and thus of what kinds of moves they are going to make in the game.
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4. Symmetry. Players have equal rights in sharing the set S. A consequence of this assumption is that, at a minimum, each player is entitled to a proportional share of S. for 2 players, each is entitled to at least one-half of S for 3 players, each is entitled to at least one-third of S. Fair Shares and Fair-Division Methods Suppose that s denotes a share of the booty S and that P is one of the players in a fair-division game with N players. We will say that s is a fair share to player P if s is worth at least 1/N th of the total value of S in the opinion of P.
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