CMU 15-896 Fair division 3: Rent division Teacher: Ariel - PowerPoint PPT Presentation

CMU 15-896 Fair division 3: Rent division Teacher: Ariel Procaccia

A true story • In 2001 I moved into an apartment in Jerusalem with Naomi and Nir • One larger bedroom, two smaller bedrooms • Naomi and I searched for the apartment, Nir was having fun in South America • Nir’s argument: I should have the large room because I had no say in choosing apartment Made sense at the time! o • How to fairly divide the rent? 15896 Spring 2016: Lecture 8 2

Sperner’s Lemma • Triangle partitioned into elementary triangles 3 • Label vertices by using 1 2 Sperner labeling: 1 2 3 Main vertices are different o 3 3 2 2 Label of vertex on an edge o ��, �� of � is � or � 1 1 2 1 2 • Lemma: Any Sperner 1 2 2 1 1 2 labeling contains at least one fully labeled elementary triangle 15896 Spring 2016: Lecture 8 3

Proof of Lemma • Doors are 12 edges • Rooms are elementary 3 triangles 1 2 1 2 3 • #doors on the boundary 3 3 2 2 of is odd 1 1 2 1 2 • Every room has 1 2 2 1 1 2 doors; one door iff the room is 123 15896 Spring 2016: Lecture 8 4

Proof of Lemma • Start at door on boundary and walk through it 3 • Room is fully labeled or it 1 2 has another door... 1 2 3 • No room visited twice 3 3 2 2 • Eventually walk into fully 1 1 2 1 2 labeled room or back to 1 2 2 1 1 2 boundary • But #doors on boundary is odd 15896 Spring 2016: Lecture 8 5

Fair rent division • Assume there are three housemates A, B, C • Goal is to divide rent so that each person wants different room • Sum of prices for three rooms is 1 • Can represent possible partitions as triangle 15896 Spring 2016: Lecture 8 6

�0,0,1� 0, 1 2 , 1 2 1 3 , 1 3 , 1 3 �1,0,0� �0,1,0� 15896 Spring 2016: Lecture 8 7

Fair rent division • “Triangulate” and assign “ownership” of each vertex to each of A, B, and C ... • ... in a way that each elementary triangle is an ABC triangle 15896 Spring 2016: Lecture 8 8

A B C C A B A B C A B C A B C C A B C A B 15896 Spring 2016: Lecture 8 9

Fair rent division • Ask the owner of each vertex to tell us which room he prefers • This gives a new labeling by 1, 2, 3 • Assume that a person wants a free room if one is offered to him 15896 Spring 2016: Lecture 8 10

• Choice of rooms on edges is constrained by the free room assumption 1 or 2 A B C C A B A B C A B C A B C 1 2 C A B C A B or 3 or 3 3 only 15896 Spring 2016: Lecture 8 11

• Sperner’s lemma (variant): such a labeling must have a 123 triangle 1 or 2 A B C C A B 2 3 A B C A 1 B C A B C 1 2 C A B C A B or 3 or 3 3 only 15896 Spring 2016: Lecture 8 12

Fair rent division • Such a triangle is nothing but an approximately envy free allocation! • By making the triangulation finer, we can increase accuracy • In the limit we obtain a completely envy free allocation • Same techniques generalize to more housemates [Su 1999] 15896 Spring 2016: Lecture 8 13

Quasi-linear utilities • Suppose each player has value �� for room • The utility of player for getting room at price � is �� � is envy free if • �� � � � �� � � � • Theorem: An envy-free solution always exists under quasi-linear utilities 15896 Spring 2016: Lecture 8 14

Quasi-linear utilities Poll 1: Suppose is an EF allocation. Then: is maximized 1. �� � � � that � There is no 2. Pareto-dominates Both 3. Neither 4. 15896 Spring 2016: Lecture 8 15

Which model is better? • Advantage of quasi-linear utilities: preference elicitation is easy Each player reports a single number in one o shot • Disadvantage of quasi-linear utilities: does not correctly model real-world situations I want the room but I really can’t spend o more than $500 on rent 15896 Spring 2016: Lecture 8 16

15896 Spring 2016: Lecture 8

Computational resources • Setting: allocating multiple homogeneous resources to agents with different requirements • Running example: shared cluster • Assumption: agents have proportional demands for their resources (Leontief preferences) • Example: Agent has requirement (2 CPU,1 RAM) for each o copy of task Indifferent between allocations (4,2) and (5,2) o 15896 Spring 2016: Lecture 8 18

Model • Set of players and set of resources • Demand of player is �� , � �� s.t. �� �� • Allocation where �� is � �� �� the fraction of allocated to • Preferences induced by the utility function � � �∈� �� �� 15896 Spring 2016: Lecture 8 19

Dominant resource fairness • Dominant resource of = s.t. �� • Dominant share of = �� for dominant • Mechanism: allocate proportionally to demands and equalize dominant shares Agent 1 alloc. Agent 2 alloc. Total alloc. 15896 Spring 2016: Lecture 8 20

Formally... • DRF finds and allocates to an �� fraction of resource r: �� �∈� � • Equivalently, �∈� ∑ � �� �∈� � � • Example: �� �� �� �� � � � � then � � � �� 15896 Spring 2016: Lecture 8 21

Axiomatic properties • Pareto optimality (PO) • Envy-freeness (EF) • Proportionality (a.k.a. sharing incentives, individual rationality): � � � • Strategyproofness (SP) 15896 Spring 2016: Lecture 8 22

Properties of DRF • An allocation � is non-wasteful if s.t. �� for all �� � • If � is non-wasteful and � � � � � then for all �� �� • Theorem [Ghodsi et al. 2011]: DRF is PO, EF, proportional, and SP 15896 Spring 2016: Lecture 8 23

Proof of theorem • PO: obvious • EF: Let be the dominant resource of o �� �� �� �� o • Proportionality: For every , �� �∈� o � � Therefore, o ∑ ��� � �� � �∈� � 15896 Spring 2016: Lecture 8 24

Proof of theorem • Strategyproofness: � are the manipulated demands; �� � �� �� o for all � � � Allocation is �� �� o � If , is the dominant resource of , o � � � � then �� �� �� �� �� � If , let be the resource saturated by o ( , then �� �∈� � � � �� � 1 � � � �� � 1 � � �� �� � 1 � � �′� �� � 1 � � � �� � � �� ��� ��� ��� ��� 15896 Spring 2016: Lecture 8 25