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

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Also ok: Second from the left, Median, etc.

Discrete facility location

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• 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?