Mechanism Design on Discrete Lines and Cycles

1. Mechanism Design on Discrete Lines and Cycles EladDokow, Michal Feldman, Reshef Meir and IlanNehama

2. 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 B 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?

6. 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.

7. 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”)

8. Main result 1 A full characterization of onto SP mechanisms on discrete lines What about cycles?

9. Non dictatorial mechanisms • Consider a small cycle (e.g. |C|=6)

10. Non dictatorial mechanisms • Take the longest arc between a pair of agents

11. Non dictatorial mechanisms • Take the longest arc between a pair of agents • Place the facility on the agent opposing the arc

12. Main result 2 Every SP and onto mechanism on (sufficiently large) cycles has a 1-dictator

13. Main result 2 Every SP and onto mechanism on (sufficiently large) cycles has a 1-dictator

14. 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)

15. 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)

16. How large are large cycles?

17. How large are large cycles? • 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

18. 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

19. The Binary cube There is a natural embedding of lines in the Binary cube

20. The Binary cube There is a natural embedding of lines in the Binary cube Also for cycles of even length

21. The Binary cube There is a natural embedding of lines in the Binary cube Also for cycles of even length

22. The Binary cube We can characterize onto SP mechanisms using properties defined w.r.t. the cube.

23. The Binary cube We can characterize onto SP mechanisms using properties defined w.r.t. the cube.

24. The Binary cube We can characterize onto SP mechanisms using properties defined w.r.t. the cube.

25. The Binary cube We can characterize onto SP mechanisms using properties defined w.r.t. the cube.

26. The Binary cube We can characterize onto SP mechanisms using properties defined w.r.t. the cube.

27. Future work • Other graph topologies • trees • Randomized mechanisms • An open question: is there a topology where every SP mechanism is a random dictator?