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FinalExamMGF1107

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

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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

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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)
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

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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.
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

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a) Suppose Nancy gets X and Sy gets Y. Is this allocation proportional? Envy-free? Equitable?
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FinalExamMGF1107 - 22:33[5 FAIR DIVISION METHODS 5.0...

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