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
Slides are courtesy of Ariel Procaccia
3
Example
• Mechanism: select the leftmost reported location
• Mechanism is strategyproof
Also ok: Second from the left, Median, etc.
A
B
C
D
E
4
Discrete facility location
• A facility cannot be placed just anywhere
• Allowed locations are vertices of a graph
(unweighted)
• Agents care about their distance from the facility
5
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
n=2
Size ≤ 12
n=3
n>3
Non-dictatorial
-
1-Dictatorial
Size ≥ 13
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
(|C|n)
8000
• Search space size is |C|
[= 20
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
Cube-monotone
(Large) even-sized cycles
Cube-monotone
The Binary cube
We can characterize onto SP mechanisms
using properties defined w.r.t. the cube.
Lines
Cube-monotone
Independent of Disjoint
Attributes (IDA)
(Large) even-sized cycles
Cube-monotone
Independent of Disjoint
Attributes (IDA)
The Binary cube
We can characterize onto SP mechanisms
using properties defined w.r.t. the cube.
Lines
Cube-monotone
Independent of Disjoint
Attributes (IDA)
(Large) even-sized cycles
Cube-monotone
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
Cube-monotone
Independent of Disjoint
Attributes (IDA)
(Large) even-sized cycles
Cube-monotone
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?