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

Network design game S i – strategy of player i is some path that connects s i to t i State S=(S 1,S 2,…,S n ) s1s1 s2s2 t2t2 t1t1

Network design game s1s1 v s2s2 t2t2 t1t1 $2 $1 $5 $3 $2 $1 $2 $8 $2 C(1) = 2 + 8/2 = 6 C(2) = 1 + 8/ = 9 cost to the player: total cost:

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

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

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

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 cost of resulting Nash equilbirium

Improvement moves r Edges in OPT Edges in graph

Improvement moves r Edges in OPT Edges in graph

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

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

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

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

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

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

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 EE, OPT, and moves

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

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

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

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

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

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

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

Scheduling phase r OPT edges graph edges x x/8 1 2 u dashed edges unused in S

Scheduling phase 1 2 r u OPT edges graph edges x x/8 dashed edges unused in S

Scheduling phase 1. Player u performs some move 2. For all players w in order of increasing dist OPT (u,w): If, Path OPT (w,u) followed by the path from u to r is better for w, then w chooses this strategy. 3. While possible, schedule OPT and EE moves

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

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

Proof sketch: S' – strategy after move Step 1: In state S', strategy of w is an improving move r u v w SwSw SvSv SuSu c(e) S' u dist OPT (u,w)<x/8

Cost of proposed strategy of w is at most Proof sketch of Step 1: r u v w SwSw SvSv SuSu c(e) S' u We show, that dist OPT (u,w) <x/8

Proof sketch of Step 1: S ince no OPT move allowed, 3. u made an improvement move, so  result follows from (2) and (3). r u v w SwSw SvSv SuSu c(e) S' u dist OPT (u,w) <x/8

Proof sketch: Step 2: It can be shown by induction, that all players will take proposed strategy dist OPT (u,w) <x/8 r u v w SwSw SvSv SuSu c(e) S' u

Let e 1 =(u 1,v 1 ), e 2 =(u 2,v 2 ) be two edges that belong to Nash, e 1  OPT and e 2  OPT. Lemma: Properties of Scheduling algorithm(2)

u2u2 r u1u1 OPT edges graph edges dashed edges unused in S e1e1 e2e2 c(e 1 )≤c(e 2 ) dist OPT (u 1,u 2 )≤c(e 1 )/8. c(e 1 )/8 dist OPT (v,w) c(e 2 )/8 Proof :

Crowded edge amortization r u c(e) = x x/8 At least logn players 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)

Light edge amortization At most logn players inside the ball of radius x v /8 Lemma: The total cost of light edges is C(OPT) ∙ loglogn

Proof: Look at Nash Equilibrium Mark light vertices

Proof: Choose vertex with maximum weight W 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

Proof: Continue the process

Proof: Draw a ball of radius W/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

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

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?