Algorithms and Economics of Networks: Coordination Mechanisms

1. Algorithms and Economics of Networks:Coordination Mechanisms Abraham Flaxman and Vahab Mirrokni, Microsoft Research

2. Price on Anarchy [KP, RT] • Selfish users • User goal: minimize its cost • Nash Equilibrium (NE) • System goal (e.g. Social welfare) • The worst ratio of NE cost to OPT cost

3. Price of Anarchy Concept • Not algorithmic • Only analysis • What to do if PoA is large • How to influence the system

4. Possible Solutions • Change the system (add tolls, payments) • Stackelberg strategy = control some users • Disadvantages: changing the settings, global knowledge • Challenge: influence within the same setting and locally (distributed)

5. Coordination Mechanism [CKN] Mechanism: local policy (algorithm) that assigns a cost for each strategy of the user • Advantages: local, same type of cost • Goal: achieving good NE • Example: scheduling jobs on machines

6. Unrelated Machine Scheduling • m unrelated machines • n jobs – each owned by different user • p(i,j) - processing time of job i on machine j • Social Objective: minimize completion time • User goal: minimize its own completion time (Makespan: Cmax)

7. Unrelated Machines Scheduling

8. Coordination Mechanism for Scheduling Policy for each machine (algorithm) which decides how to schedule jobs assigned to it Each Policy induces NE on jobs

9. Local Scheduling Policies

10. Local Scheduling Policies • Shortest-first Policy • Longest-first Policy • Random Order Policy • MakeSpan Policy

11. Challenge Design policies that results in good NE (i.e. low PoA)

12. Unrelated Machines Scheduling

13. Equilibrium for Longest First

14. PoA of Longest First • Results in poor NE • The PoA is unbounded even for 2 machines • The optimum completion time is low • The completion time of NE is large

15. Machine Scheduling Models • Identical Machines: P||Cmax • Related Machines: Q||Cmax (Different Speeds) • Restricted Assignment: B||Cmax • Unrelated Machines: R||Cmax

16. PoA Results

17. Pure NE Results

18. Type of Policies • Local policy – depends on jobs assigned to machine • Strongly local policy - depends only on processing time of jobs on that machine • Ordering Policy = IIA (independence of irrelevant alternative)

19. Lower Bound for Strongly Local Policy • We start with Shortest-First • Extend it to arbitrary strongly local IIA policy • Shortest-First is interesting by its own

20. Shortest-First • Approx factor known to be at most m • NE can be computed by shortest-first greedy algorithm (Alg D by Ibarra and Kim) • An open question from 1977 • We show it is at least m/2

21. Idea of the Proof • m types of jobs • Type j can be scheduled on machines j & j+1 • Processing time of type j on machine j is low and on machine j+1 is high (ratio is j) • All jobs on machine j have almost the same processing time

22. Example for Shortest-First

23. Idea of the Proof • OPT assign all jobs of type j to machine j • Number of jobs is chosen such that OPT has the same completion time for all machines

24. Optimal Assignment

25. Idea of the Proof • In NE about half jobs of type j are on machine j and half on machine j+1 • Completion time of NE grows linearly in m

26. Equilibrium for Shortest-First

27. Extend to Arbitrary Strongly Local • Structure is similar to lower bound for Shortest-First • Arbitrary ordering function is given for each machine • Indices of jobs are chosen to behave similar to the above example

28. Efficiency Based Algorithm • Order jobs on each machine by their efficiency • Efficiency of job on machine is: The ratio between job’s best processing time to its processing time on this machine • PoA of algorithm is O(log m)

29. Equilibrium Improves

30. Efficiency Based Algorithm • Unfortunately – pure NE may not exist • Iterative improvement may cycle • Modified algorithm guarantees convergence and pure NE with PoA of O(log^2 m)

31. Modified Algorithm • Each machine simulate log m submachines (by round robin) • Submachine k of machine j handles jobs on efficiency between 2^{-k} and 2^{-k+1} • Jobs are ordered on submachine by Shortest-First • PoA of algorithm is O(log^2 m)

32. Summary Coordination Mechanism: • Influence on the quality of the equilibrium Unrelated Machines: • m – lower bound • Shortest-First is at least m • Local order by efficiency O(log m) – optimal • Pure + Convergence O(log^2 m)

33. Discussion and Open Problems Non ordering strategies – get below log m • Extend to network routing • Show more effective usage of coordination mechanism