FinalExamMGF1107

FinalExamMGF1107 - 22:33 [5] FAIR DIVISION METHODS 5.0]...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
22:33 [5] FAIR DIVISION METHODS 5.0] Explain what Fair Division is. Collection of mathematical methods (algorithms) with the objective of dividing a group of objects (divisible or indivisible) among a set of individuals in such a way that each of the recipients is satisfied with the shares that they receive. 5.1] Describe the cases of Fair Division: Discrete, Continuous, Mixed Cases Discrete Cases: The objects to be distributed are indivisible (houses, vehicles, art, etc.) Continous Cases: Objects are divisible into infinitely many parts. (cakes, money, land…) Mixed cases: Part of objects are divisible objects and others are indivisible. [5.2] Desirable Properties of Fair Division Methods [5.2.1] Proportionality
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A fair division method is said to be proportional if each individual beliees (according to their own perception) that they received at least 1/n (one- n th) of the whole. Ex: divide a cake among 3 individuals. A division method is proportional if each individual believes that they received at least 1/3 (one-third) of the whole thing. [5.2.2] Envy-Freeness A fair division method is envy-free if each individual believes that her/his share is at least as large (or as desirable) as the share of any other individual (this way no one will be envious of another person’s share) REMARK: Every method that is envy-free is automatically proportional [5.2.3] Equitability A fair division method is said to be equitable if each individual believes (according to their own perception) that every recipient received an equal amount. [5.2.4] Pareto Optimality ( or Efficiency)
Background image of page 2
A fair division method is said to be pareto optimal if there is no other allocation that will make one individual better off without making someone else worse off. [5.3] EXAMPLES 5.3.1) Suppose that four items (W, X, Y, Z) must be divided fairly among four people (Nancy, Amal, Sy, Ted). Each player is allowed to spread 100 points over the four items to indicate the worth (the importance) of each item to that player. The distribution is as follows: Ite m Nan cy Am al Sy Te d W 25 50 30 25 X 25 30 20 25 Y 25 10 20 30 Z 25 10 30 20
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
a) Suppose that Nancy gets X, Amal gets W, Sy gets Z, and Ted gets Y. Is this allocation proportional? Envy-free? Equitable? Pareto optimal? In each case, explain why. Proportional because each individual believes they received minimum 25% of the whole. Envy-Free since each individual believes that they received at least one-fourth of the whole, nobody will be envious of another person’s share. Equitable since Nancy believes that she received 25% and Amal believes that she received 50% then the allocation is not equitable b) Suppose that Nancy gets Y, Amal gets Z, Sy gets W, and Ted gets X. Is this allocation proportional? Envy-free? Equitable? Pareto optimal? In each case, explain why.
Background image of page 4
5.3.2) Suppose that two items (X, Y) must be divided fairly among Nancy and Sy. Each player is allowed to spread 100 points over the two items to indicate the worth (the importance) of each item to that player. The distribution is as follows: Ite m Nanc y Sy X 55 45 Y 45 55
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
a) Suppose Nancy gets X and Sy gets Y. Is this allocation proportional? Envy-free? Equitable?
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/08/2011 for the course MGF 1107 taught by Professor Storfer during the Spring '08 term at FIU.

Page1 / 39

FinalExamMGF1107 - 22:33 [5] FAIR DIVISION METHODS 5.0]...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online