mechanism design on discrete lines and cycles elad dokow, michal feldman, reshef meir and ilan...
Post on 04-Jan-2016
216 Views
Preview:
TRANSCRIPT
Mechanism Design on Discrete Lines and Cycles
Elad Dokow, Michal Feldman, Reshef Meir and Ilan Nehama
Example
• Suppose we have two agents, A and B• Mechanism: take the average A mechanism is strategyproof if agents can never
benefit from lying = the distance from their location cannot decrease by misreporting it
3Slides are courtesy of Ariel Procaccia
Example
BB EECC DDAA BB
• Mechanism: select the leftmost reported location• Mechanism is strategyproof
BB
4
Also ok: Second from the left, Median, etc.
Discrete facility location
5
• A facility cannot be placed just anywhere• Allowed locations are vertices of a graph
(unweighted)• Agents care about their distance from the facility
Main questions
Given a graph G, characterize all deterministic strategyproof (SP) mechanisms on G
Are there SP mechanisms with good social welfare?
Previous work
• Schummer and Vohra 2004:
Full characterization on continuous Lines, Cycles and Trees.– On every continuous cycle there is a dictator
• Alon et al. 2010: – optimal welfare on (cont.) Trees
– Ω(n) approximation on cyclic graphs
– Randomized mechanisms
• Moulin 1980: Single-peaked preferences.
Notations
• Denote x = f(a) = f(a1,a2,…,an)
• d(x,y) is the distance between x and y
• A k-dictator is an agent that is always at distance (at most) k from the facility, i.e.
d(ai,f(a)) ≤ k for all a
• A mechanism is anonymous if it treats all agents symmetrically (“fairly”)
Main result 1
A full characterization of onto SP mechanisms on discrete lines
What about cycles?
Non dictatorial mechanisms
• Consider a small cycle (e.g. |C|=6)
Non dictatorial mechanisms
• Take the longest arc between a pair of agents
Non dictatorial mechanisms
• Take the longest arc between a pair of agents• Place the facility on the agent opposing the arc
Main result 2
Every SP and onto mechanism on (sufficiently large) cycles has a 1-dictator
Main result 2
Every SP and onto mechanism on (sufficiently large) cycles has a 1-dictator
Proof outline
• The case of two agents:– Every SP and onto mechanism is unanimous
– “ “ “ “ is Pareto
– The facility must be next to some agent
– It is always the same agent (the 1-dictator)
Proof outline (cont.)
• For three agents:– Either (a) there is a 1-dictator, or (b) every pair is a
“dictator” when in the same place
– For large cycles, (b) is impossible
– Thus there is a 1-dictator
• For n>3 agents:– A reduction to n-1 agents (similar to SV’04)
How large are large cycles?
# of agents Anonymous Non-dictatorial 1-Dictatorial
n = 2 Size ≤ 12 - Size ≥ 13
n = 3
n > 3
How large are large cycles?
# of agents Anonymous Non-dictatorial 1-Dictatorial
n = 2 Size ≤ 12 - Size ≥ 13
n = 3 Size ≤ 14 (and 16) - Size ≥ 17 (and 15)
n > 3 Impossible if size>n Size ≤ 14 (and 16) Size ≥ 17 (and 15)
• Our proof only works for size ≥ 22
• For smaller cycles – used exhaustive search
• Search space size is |C|(|C|n) [= 208000 for |C|=20]
…but we can narrow it significantly
Implications
• Graphs with several cycles
• A lower bound on the social cost
• A simpler proof for the continuous case
• Applications for Judgment aggregation and Binary classification
The Binary cube
There is a natural embedding of lines in the Binary cube
The Binary cube
There is a natural embedding of lines in the Binary cube
Also for cycles of even length
The Binary cube
There is a natural embedding of lines in the Binary cube
Also for cycles of even length
The Binary cube
We can characterize onto SP mechanisms using properties defined w.r.t. the cube.
Lines (Large) even-sized cycles
The Binary cube
We can characterize onto SP mechanisms using properties defined w.r.t. the cube.
Lines (Large) even-sized cyclesCube-monotone Cube-monotone
The Binary cube
We can characterize onto SP mechanisms using properties defined w.r.t. the cube.
Lines (Large) even-sized cyclesCube-monotone Cube-monotoneIndependent of Disjoint Attributes (IDA)
Independent of Disjoint Attributes (IDA)
The Binary cube
We can characterize onto SP mechanisms using properties defined w.r.t. the cube.
Lines (Large) even-sized cyclesCube-monotone Cube-monotoneIndependent of Disjoint Attributes (IDA)
Independent of Disjoint Attributes (IDA)1-Dictatorial
The Binary cube
We can characterize onto SP mechanisms using properties defined w.r.t. the cube.
Lines (Large) even-sized cyclesCube-monotone Cube-monotoneIndependent of Disjoint Attributes (IDA)
Independent of Disjoint Attributes (IDA)
1-IIA 1-Dictatorial
Future work
• Other graph topologies– trees
• Randomized mechanisms– An open question: is there a topology where every
SP mechanism is a random dictator?
top related