CMU 15-896 Mechanism design 1: Without money Teacher: Ariel Procaccia
Approximate MD wo money • We saw in kidney exchange that the optimal solution may not be strategyproof • Approximation can be a way to quantify how much we sacrifice by insisting on strategyproofness (Example: Mix and Match) 15896 Spring 2016: Lecture 21 2
Facility location • Each player has a location � • Given � , choose a facility � location • � � • Two objective functions Social cost: sc � � o Maximum cost: mc � o � • Social cost: the median is optimal and SP 15896 Spring 2016: Lecture 21 3
The median is SP 15896 Spring 2016: Lecture 21 4
MC + det • What about maximum cost as the objective? Poll 1: What is the approximation ratio of the median to the max cost? 1. 2. 3. 4. 15896 Spring 2016: Lecture 21 5
MC + det • Theorem [P and Tennenholtz 2009]: No deterministic SP mechanism has an approximation ration to the max cost • Proof: 15896 Spring 2016: Lecture 21 6
MC + rand • The Left-Right-Middle (LRM) Mechanism: Choose � with prob. , � with prob. , and their average with prob. Poll 2: What is the approximation ratio of the LRM Mechanism to the max cost? 5/4 1. 3/2 2. 7/4 3. 2 4. 15896 Spring 2016: Lecture 21 7
MC + rand • Theorem [P and Tennenholtz 2009]: LRM is SP • Proof: 1/4 1/2 1/4 2� � 1/2 1/4 1/4 15896 Spring 2016: Lecture 21 8
MC + rand • Theorem [P and Tennenholtz 2009]: No randomized SP mechanism has an approximation ratio • Proof: � � � 0, � � � 1 , � � � � o cost �, � � � cost �, � � � 1; w log cost �, � � � 1/2 o � � 2 � � � 0, � � o By SP, the expected distance from � � is at least ½ o Expected max cost at least 3/2 , because for every o � ∈ � , the expected cost is � � 1 � 1 ∎ 15896 Spring 2016: Lecture 21 9
From lines to circles • Continuous circle is the distance on the circle • • Assume that the circumference is • “Applications”: Telecommunications o network with ring topology Scheduling a daily task o 15896 Spring 2016: Lecture 21 10
MC + rand + circle • Semicircle like an interval on a line • If all agents are on 1/4 one semicircle, can apply LRM • Problematic otherwise 1/4 15896 Spring 2016: Lecture 21 11
MC + rand + circle • Random Point (RP) Mechanism: Choose a random point on the circle • Obviously horrible if players are close together • Gives a approx if the players cannot be placed on one semicircle Worst case: many agents uniformly distributed over o slightly more than a semicircle If the mechanism chooses a point outside the o semicircle (prob. 1/2 ), exp. max cost is roughly 1/2 If the mechanism chooses a point inside the o semicircle (prob. 1/2 ), exp. max cost is roughly 3/8 15896 Spring 2016: Lecture 21 12
MC + rand + circle • Hybrid Mechanism 1: Use LRM if players are on one semicircle, RP if not • Gives a approx • Surprisingly, Hybrid Mechanism 1 is also SP! 15896 Spring 2016: Lecture 21 13
Hybrid Mechanism 1 is SP � � • Deviation where RP or LRM is used before and after is not beneficial • LRM to RP: expected cost of is at � ℓ most before, exactly after; focus on RP to LRM � � and are extreme locations in new • profile, and their antipodal � ℓ �̂ points � ℓ • Because agents were not on one � � � semicircle in , � 15896 Spring 2016: Lecture 21 14
Hybrid Mechanism 1 is SP � � = center of • � ℓ �̂ , because • � � ℓ , , and � � � � • Hence, � cost lrm �′ , � � � 1 4 � � � , ℓ � 1 4 � � � , � � 1 2 ��� � , �� � 1 � 1 2 ⋅ 1 4 � � � , ℓ � � � � , � 4 � 1 4 � cost�rp � , � � � ∎ 15896 Spring 2016: Lecture 21 15
MC + rand + circle • Goal: improve the approx ratio of Hybrid 1? • Random Midpoint (RM) Mechanism: choose midpoint of arc between two antipodal points with prob. proportional to length 15896 Spring 2016: Lecture 21 16
MC + rand + circle Poll 3: The worst example you can think of for RM gives a ratio of what to the max cost? 1. 2. 3. 4. 15896 Spring 2016: Lecture 21 17
MC + rand + circle � • Lemma: When the players are not on a semicircle, RM gives a approx • Proof: � � � � � � length of the longest arc between o two adjacent players, w.l.o.g. � � and � � � � 1/2 because otherwise players are on one semicircle o Opt � at center of � � � and � � � , so OPT � �1 � ��/2 o RM selects � with probability � , and a solution with o cost at most 1/2 with prob. 1 � � � ��� � � ��� � � � 1 � � � � ∎ ��� o � 15896 Spring 2016: Lecture 21 18
MC + rand + circle • Hybrid Mechanism 2: Use LRM if players are on one semicircle, RM if not • Theorem [Alon et al., 2010]: Hybrid Mechanism 2 is SP and gives a approx to the max cost • The proof of SP is a rather tedious case analysis… but the fact that it’s SP is quite amazing! 15896 Spring 2016: Lecture 21 19
MC + rand + • Let’s go back to the line, but now there are facilities • For � � � � � � • Optimal solution for max cost: cover with intervals of length in a way that minimizes ; place the th facility in the center of the th interval 15896 Spring 2016: Lecture 21 20
MC + rand + • Equal Cost (EC) Mechanism: Cover � with � intervals as above o With prob. 1/2 , choose the leftmost (resp., o righmost) point of every odd interval, and the rightmost (resp., leftmost) point of every even interval • Theorem [Fotakis and Tzamos 2013]: EC is an SP 2-approximation mechanism for the max cost 15896 Spring 2016: Lecture 21 21
overview [Fotakis and Tzamos 2013] 15896 Spring 2016: Lecture 21 22
Recommend
More recommend
