Best Response Dynamics in Multicast Cost Sharing

1. Best Response Dynamics in Multicast Cost Sharing Seffi Naor Microsoft Research and Technion Based on papers with: C. Chekuri, J. Chuzhoy, L. Lewin-Eytan, and A. Orda [EC 2006] M. Charikar, C. Mattieu, H. Karloff, M. Saks [2007] TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

2. Motivation • Traditional networks – single entity, single control objective. • Modern networking – many entities, different parties. • Users act selfishly, maximizing their objective function. • Decisions of each user are based on the state of the network, which depends on the behavior of the other users.  non-cooperative network games.

3. Our Framework • A network shared by a finite number of users. • Each edge has a fixed cost. • Cost sharing method defines the rules of the game: determines the mutual influence between players. • Performance of a user is total payment = sum of payments for all the edges it uses. • Two fundamental models: • Thecongestion model. • Thecost sharing model.

4. In both models: • Each user routes its traffic over a minimum-cost path. • Splittable routing model vs. unsplittablerouting model.

5. The Multicast Game • A special root node r, and a set of nreceivers (players). • A player’s strategy isa routing decision – the choice of a single path to r. • Egalitarian(Shapley( cost sharing mechanism: the cost of each edge is evenly split among the players using it.Each player on edge e with ne players pays: ce / ne • The goal of the players is to connect to the root by making a routing decision minimizing their payment.

6. The Multicast Game Two different models: • The integral model: each player connects to the root through a single path. • The fractional model: each player is allowed to split its connection to the root into several paths (fractions add up to 1).

7. Nash Equilibrium • Players are rational. • Each player knows the rules of the underlying game. Nash Equilibrium:no player can unilaterally improve its cost by changing its path to the root. Cost of a path takes into account cost sharing. • Nash equilibrium solutions are stable operating points.

8. The Price of Anarchy • Nash equilibrium outcomes do not necessarily optimize the overall network performance. Price of Anarchy:The ratio between the cost of the worst Nash equilibrium and the (social) optimum. • Quantifies the “penalty” incurred by lack of cooperation.

9. The Integral Multicast Game • Potential function Φ of a solution T [Rosenthal `73]: • Exact potential:changein cost of a connection of player i to the root is equal to thechangein the potentialΦ. • If edge e is deleted from T: Φ=Φ - ce / ne(T) • If edge e is added to T: Φ=Φ + ce / (ne(T)+1)

10. The Integral Multicast Game • Finite strategy space  Φ has an optimal value. • Φ  Nash equilibrium existence. • Global / Local optima of Φ correspond to a NE. • A Nash equilibrium solution is a tree rooted at r spanning the players. • Special case of a congestion game.

11. Price of Anarchy vs. Price of Stability • Price of anarchycan be as bad as (n). • Price of stability– ratio between the cost of best Nash solution to the cost of OPT. • Outcome of scenarios in the ‘middle ground’ between centrally enforced solutions and non-cooperative games. • E.g.: central entity can enforce the initial operating point. • [Anshelevich et al., FOCS 2004] • Directed graphs - price of stability is θ(log n). • Undirected graphs – upper bound on the price of stability is O(log n). PoS can be reached from a 2-approximate Steiner tree configuration: C(TNash)  Φ(TNash)  Φ(T2-Seiner)  log n ∙C (T2-Steiner) r t

12. Best Response Dynamics • Each player, in its turn, selects a path minimizing its cost (best response). • Eventually, an equilibrium point is reached. • PoA strongly depends on the choice of the initial configuration. • Starting from a near-optimal solution may be hard to enforce: requires relying on a central trusted authority. Question: What happens if we start from an emptyconfiguration?[Chekuri, Chuzhoy, Lewin-Eytan, Naor, and Orda, EC 2006] • Round 1: Players arrive one by one, each player plays best response. • Round 2: Best response dynamics continue in arbitrary order till NE. Question: Can a good equilibrium always be achieved as a consequence of best-response dynamics in this model?

13. Cost of user 1:c (r, x, 1) =1+ε c (r , 1) =1 r r r r r r r r r r Greedy cost of 3, … ,n = 1 Cost of user 2:c(r , x, 2) = 1+ε c(r, 1, x, 2 ) = 1+2ε c (r, 2) = 1 1 1 3/4 1 1 1 1 x x x x x x x x x x x x x x x ¼ + ε ¼ + ε ¼ + ε ¼ + ε … 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 n n-2 n-1 Price of anarchy = 4 Can a good equilibrium be achieved as a consequence of best-response dynamics?

14. Results • The integral multicast game for undirected graphs: • Upper bound of O(log3n) on the PoA of best-response dynamics in the two-roundgame starting from an “empty” configuration. (Improving over the bound of [CCLNO-EC06] of.) • Upper bound of O(log2+n) on the cost of the solution at the end of the first round. • Lower bound of (log n) on the PoA of this game. • Computing a Nash equilibrium minimizing Rosenthal’s potential function is NP-hard.

15. Price of Anarchy: Lower Bound Theorem:Price of anarchy of our game ¸(logn). Proof:Adaptation of lower bound proof for the online Steiner problem [Imaze&Waxman] • Online Steiner problem: used edges have cost 0 • Take hard instance [IW] and replace each terminal by a star of n2 + 1 terminals at zero distance • The n2 + 1 terminals choose the same path to root cost of used edge becomes negligible

16. PoA in Undirected Graphs: Upper Bound Analysis is performed in two steps: • Round 1: players connect one by one to the root via best response. • Solution T is reached after a sequence of arrivals t1 , t2 ,…, tn. We show:  O(log3n )∙ c(TOPT ) c(T) ·O(log2+n )∙ c(TOPT ) • Round 2: players play in arbitrary order till NE is reached.c(TNash)  Φ(TNash)  Φ(T)

17. The First Round • Choose a threshold 2 (0,1). • Terminal t is -good if cost of next terminal using the same path as t· (1-) ¢ cost(t). • Otherwise, terminal t is -bad. • Idea: bound separately the contribution to Φ of the -good terminals and the -bad terminals.

18. t d t’ Charging -good Terminals r • Suppose there is a tree T’ spanning the -good terminals. • Upon arrival t pays: cost(t) • Upon arrival, t’ pays at most: cost(t’)  d + (1-) ¢ cost(t) cost(t) • t and t’ are-goodterminals • t arrives first, then t’ arrives

19. Charging -good Terminals (contd.) r Charges decay at an exponential rate along a root – leaf path in T’ • Upon arrival of tiit pays at most: cost(ti) ·di + (1-)¢cost(ti-1) • cost(tk) · dk + dk-1(1-) + dk-2(1-)2 + … + d1(1-)k-1 d1 t1 d2 t2 d3 t3 d4 t4 dk tk The charge to each edge e2T’: ·d(e) ¢i(1-)i¢ne(i) • t1,…, tkare-goodterminals • arrival order: t1,…, tk

20. Auxiliary Tree TOPT is transformed to an auxiliary treeT’defined on the set of -good terminals: • The descendants of terminal t in T’ are terminals which have arrived after t. • c(T’) · O(1/¢ logn) ¢c(TOPT ) • The depth ofT’ ·O(1/¢ logn)

21. Contribution of -good Terminals Theorem:The contribution of the -good terminals to Φ in the first phase is bounded as follows: Proof: Follows from the properties of T’ together with the exponential decay of the charges to the edges of T’ of the -good terminals.

22. Contribution of -bad Terminals • Theorem:The contribution of the -bad terminals to Φ in the first phase is bounded as follows: • Intuition: The cost of the edges “opened” for the first time by -bad terminals constitutes only a small part of the sum of the costs of the -bad paths. • Setting  = O(1/logn) O(log4n ) upper bound on PoA • Setting  cleverly O(log3n ) upper bound on PoA

23. The Fractional Multicast Game • Players split their connection to the source. • A splittable multicast model. r • Flow on (r, y): • ¼ unit of flow is shared by users 1 & 2. • ½ unit of flow is used only by user 2. • Flow on (r, x): • ¼ unit of flow is shared by users 1 & 2. • ½ unit of flow is used only by user 1. 3/4 3/4 1/4 1/4 x y 1 2

24. The Fractional Multicast Game • The cost of each flow fraction is split evenly between its users. • ce·fe,n_e is the total cost of edge e. fe,1= 1/4 fe,2= 3/4 fe,3 = 7/8 1/4 3/4 7/8 Total cost of user 3: ce· (1/12 + 1/4 + 1/8).

25. The Fractional Multicast Game • The potential function Φ of the fractional model: • Φ is an exact potential. • A fractional flow configuration defining a local minimum of Φ corresponds to a NE.

26. Results: Fractional Multicast Game • Nash equilibrium existence. • NE - minimizing the potential function: • Can be computed in polynomial time (using LP). • It is NP-hard in the case of an integral Nash. • PoA of the computed NE is O(log n).

27. Computing a Minimum Potential NE • Create a new graph G’ = (V, E’) by replacing each edge e by n copies e1, e2, …, en. • Copy ej : “should” be used if j players use edge e. • The cost of a unit flow on copy j of edge e is ce/ j. • The undirected graph is replaced by a directed flow network.

28. Computing a Minimum Potential NE The linear program: • Objective function = potential function: • The capacity of edge ej is 0  xe_j 1. • Variables of the LP: • Flows of the users on the edges ejE’. • Capacities of the edges in E’. Constraints: • Non-aggregating flow constraint: (flow of user i on ej )  xe_j . • Aggregating flow constraint: (total flow on ej ) = j ∙ xe_j.

29. The Linear Program Theorem 5:There exists an optimal solution to the linear program that corresponds to a fractional multicast flow. • Heavily depends on the non-increasing property of the cost function. • LP can be used: • For any cost sharing mechanism that is cross-monotonic. • In case players are not restricted to have a common source. • PoA of a minimum potential fractional NE is O(log n).

30. Integral vs. Fractional Potential Minimization • There exists an instance where the gap between the integral and fractional minimum potential solutions is a small constant. • Finding an integral Nash equilibrium that minimizes the potential function is NP-hard. • Building block: a variation of the Lund-Yannakakis hardness proof of approximating the set cover problem.

31. The Weighted Multicast Game • Each player i has a positive weight wi. • The cost share of each player is proportional to its weight. • Integral: cost share of player i = ce· (wi / We) (We – weight of players currently using e) • Fractional: weighted sharing on each fraction. Theorem:A NE always exists for the weighted fractional model. Note: NE does not necessarily exists for the weighted integral model [Chen-Roughgarden, SPAA 06].

32. Thank You!