On the Price of Stability for Designing Undirected Networks with Fair Cost Allocations

1. On the Price of Stability for Designing Undirected Networks withFair Cost Allocations Svetlana Olonetsky Joint work with Amos Fiat, Haim Kaplan, Meital Levy, Ronen Shabo

2. Network design game • Si– strategy of player i is some path that connects sito ti • State S=(S1,S2,…,Sn) t1 t2 s1 s2

3. Network design game • cost to the player: • total cost: $2 C(1) = 2 + 8/2 = 6 C(2) = 1 + 8/2 + 3 +1 = 9 $3 t1 $2 t2 $1 $8 $5 $2 $2 v $2 $2 s1 $1 s2

4. Definitions • Nash Equilibrium: State S is a Nash equilibrium if for every state S′=(S1,…,Si-1, S′i, Si+1,…,Sn) • Price of stability: C(best NE) C(OPT) (Min cost Steiner forest)

5. Summary • Known Results Price of stability on directed graphs: (log n) “The Price of Stability for Network Design with Fair Cost Allocation “ [E.Anshelevich, A.Dasgupta, J.Kleinberg, E.Tardos, T. Roughgarden ] • Open problem: Price of stability on undirected graphs

6. Our results • Undirected graphs: • Common target vertexr(Multicast) • Player at every vertex • Theorem: The Price of Stability for this game is O(loglog n).

7. Proof overview • Start with OPT tree (OPT is some MST) • Describe algorithm that produces a particular sequence of improvement moves leading to Nash equilbirium • Bound costof resulting Nash equilbirium

8. Improvement moves Edges in graph r Edges in OPT

9. Improvement moves Edges in graph r Edges in OPT

10. Edges in OPT - change of strategy v previous new EE move – use Existing Edges r v no new edges were added by v

11. Edges in OPT - change of strategy v previous new OPT move – use edges in MST r v new OPT edge was added

12. move Edges in OPT - change of strategy w r previous new w new edge, not OPT, not EE, first on path from w

13. EE, OPT, and moves Lemma 1: If no EE moves possible S is a tree Proof: ≤ r v u

14. EE, OPT, and moves Lemma 2: If no OPTmoves possible  - calculated in similar way as CS(w), except that additional player counted on path from wto LCAS(v,w). Proof: • If S' differs from S by strategy of v, only edges on path from w to LCAS(v,w) can become cheaper for w. • If Lemma doesn’t hold, connect v to w and continue with w

15. EE, OPT, and moves Lemma 3 (without proof): If no EE, OPT, or moves possible  state S is in Nash equilibrium

16. EE, OPT, and moves • EE moves do not increase the total cost • OPT moves increase the Price of Stability by a factor ≤ 2 • moves can increase the total cost • Every move adds one new edge to S

17. Scheduling algorithm • The scheduler works in phases • In the beginning of a phase no OPT or EE moves are possible.

18. Scheduling phase OPT edges r graph edges dashed edges unused in S

19. Scheduling phase OPT edges r graph edges dashed edges unused in S u

20. Scheduling phase OPT edges r graph edges dashed edges unused in S x u u performs move

21. Scheduling phase OPT edges r graph edges dashed edges unused in S x u 1 loop on distOPT(u,w)

22. Scheduling phase OPT edges r graph edges dashed edges unused in S x u 1 2 loop on distOPT(u,w)

23. Scheduling phase OPT edges r graph edges unused edge dashed edges unused in S 5 3 x 6 u 1 2 loop on distOPT(u,w) unused edge 4

24. Scheduling phase OPT edges r graph edges dashed edges unused in S 5 3 x 6 u x/8 1 2 4

25. Scheduling phase OPT edges r graph edges dashed edges unused in S 5 3 x 6 u x/8 1 2 4

26. Scheduling phase • Player u performs some move • For all players w in order of increasing distOPT(u,w): • If, PathOPT(w,u) followed by the path from u to r is better for w, then w chooses this strategy. • While possible, schedule OPT and EE moves

27. Potential function This game has an exact potential function: If user i changes its strategy from Si to S′i:  

28. Properties of Scheduling algorithm(1) r Sv • Let e=(u,v), e OPT, added to S by an move Lemma: During the remainder of the phase • All users w within distOPT(u,w) ≤c(e)/8 modify theirstrategy to include u… r as the tail of their strategy. • After each move potential drops by a constant fraction of c(e) v Sw S'u c(e) Su u w distOPT(u,w)<x/8

29. Proof sketch: r Sv • S'– strategy aftermove Step 1: In state S', strategy of w is an improving move v Sw S'u c(e) Su u w distOPT(u,w)<x/8

30. Proof sketch of Step 1: Cost of proposed strategy of w is at most r We show, that Sv v Sw S'u c(e) Su u w distOPT(u,w) <x/8

31. Proof sketch of Step 1: • Since no OPT move allowed, • u made an improvement move, so  result follows from (2) and (3). r Sv v Sw S'u c(e) Su u w distOPT(u,w) <x/8

32. Proof sketch: r Sv Step 2:It can be shown by induction, that all players will take proposed strategy v Sw S'u c(e) Su u w distOPT(u,w) <x/8

33. Properties of Scheduling algorithm(2) • Let e1=(u1,v1),e2=(u2,v2) be two edges that belong to Nash, e1 OPT and e2 OPT. Lemma:

34. Proof : OPT edges r graph edges dashed edges unused in S c(e2)/8 e2 e1 u2 u1 c(e1)/8 distOPT(v,w) c(e1)≤c(e2)distOPT(u1,u2)≤c(e1)/8.

35. r 5 3 c(e) = x 6 u x/8 1 2 4 Crowded edge amortization • At least lognplayers inside the ball • Moves of players inside the ball dropped the potential by (x ∙ logn) • Initial potential value is at most C(OPT) ∙logn Lemma: The total cost of crowded edges is C(OPT)

36. Light edge amortization • At most lognplayers inside the ball of radius xv/8 • Lemma: • The total cost of light edges is C(OPT) ∙ loglogn

37. Proof: • Look at Nash Equilibrium • Mark light vertices 10 3 1 3

38. Proof: • Choose vertex with maximum weightW and draw a ball with radius W/8 • Remove light vertices inside this ball with weight less then W / log n • Total cost of removed vertices at most W 10 10 3 1 3

39. Proof: • Continue the process 10 3 3

40. Proof: • Draw a ball of radiusW/24 around remained vertices • Every point of tree can be covered by balls with radiuses: • max W / log n < R < max W • Radius size decreases by at least factor 2 • every point of tree can be covered by loglogn balls 10 3 3

41. Summary • Total cost of crowded edges: • C(OPT) • Total cost of light edges: • C(OPT) · loglogn • Price of Stability: loglogn

42. Open problems • We believe that the price of stability for this version is constant. • Can our result be applied to a single source setting where there may not be an agent in every node? • Generalization to the case where agents want to connect to different sources?