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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.: AA A A A A A A

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2 Motivation Traditional networks – single entity, single control objective. Traditional networks – single entity, single control objective. Modern networking – many entities, different parties. Modern networking – many entities, different parties. Users act selfishly, maximizing their objective function. 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. 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. non-cooperative network games.

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

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4 Congestion Model Cost Sharing Model Common in unicast routing. Common in unicast routing. Edge cost: Edge cost: Modeled by a load dependent function. Modeled by a load dependent function. Non-decreasing in the total flow of the edge. Non-decreasing in the total flow of the edge. Each user has a negative effect on the performance of other users. Each user has a negative effect on the performance of other users. Common in multicast routing. Common in multicast routing. Edge cost: Edge cost: Fixed cost Fixed cost Cost sharing mechanism determines how the cost is shared by the users. Cost sharing mechanism determines how the cost is shared by the users. Each user has a favorable effect on the performance of other users (cross monotonicty). Each user has a favorable effect on the performance of other users (cross monotonicty). In both models: Each user routes its traffic over a minimum-cost path. Each user routes its traffic over a minimum-cost path. Splittable routing model vs. unsplittable routing model. Splittable routing model vs. unsplittable routing model.

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

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

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7 Nash Equilibrium Players are rational. Players are rational. Each player knows the rules of the underlying game. 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. Nash equilibrium solutions are stable operating points.

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8 The Price of Anarchy Nash equilibrium outcomes do not necessarily optimize the overall network performance. 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. Quantifies the penalty incurred by lack of cooperation.

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

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

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11 Price of Anarchy vs. Price of Stability Price of anarchy can be as bad as (n). Price of anarchy can be as bad as (n). Price of stability – ratio between the cost of best Nash solution to the cost of OPT. 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. 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. E.g.: central entity can enforce the initial operating point. [Anshelevich et al., FOCS 2004] [Anshelevich et al., FOCS 2004] Directed graphs - price of stability is θ(log n). 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: 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(T Nash ) Φ(T Nash ) Φ(T 2-Seiner ) log n C (T 2-Steiner ) rt

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12 Each player, in its turn, selects a path minimizing its cost (best response). Each player, in its turn, selects a path minimizing its cost (best response). Eventually, an equilibrium point is reached. Eventually, an equilibrium point is reached. PoA strongly depends on the choice of the initial configuration. 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. 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 empty configuration? [Chekuri, Chuzhoy, Lewin-Eytan, Naor, and Orda, EC 2006] Round 1: Players arrive one by one, each player plays best response. Round 1: Players arrive one by one, each player plays best response. Round 2: Best response dynamics continue in arbitrary order till NE. 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? Best Response Dynamics

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13 r 213n … ¼ + ε 3/4 11 x r 1 x r 1 x r 1 x r 1 x 2 r 1 x 2 r 1 x 2 r 1 x 2 r 1 x 32 r 1 x 321 x 321 x 321 x 321 x 321 x Cost of user 1: c (r, x, 1) = 1+ε c (r, 1) = 1 Cost of user 2: c ( r, x, 2 ) = 1+ε c (r, 1, x, 2 ) = 1+2ε c (r, 2) = 1 Greedy cost of 3, …,n = 1 Price of anarchy = 4 Can a good equilibrium be achieved as a consequence of best-response dynamics? of best-response dynamics? n-2n-1

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14 Results The integral multicast game for undirected graphs: The integral multicast game for undirected graphs: Upper bound of O(log 3 n) on the PoA of best-response dynamics in the two-round game starting from an empty configuration. (Improving over the bound of [CCLNO-EC06] of.) Upper bound of O(log 3 n) on the PoA of best-response dynamics in the two-round game starting from an empty configuration. (Improving over the bound of [CCLNO-EC06] of.) Upper bound of O(log 2+ n) on the cost of the solution at the end of the first round. Upper bound of O(log 2+ 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. Lower bound of (log n) on the PoA of this game. Computing a Nash equilibrium minimizing Rosenthals potential function is NP-hard. Computing a Nash equilibrium minimizing Rosenthals potential function is NP-hard.

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15 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 Online Steiner problem: used edges have cost 0 Take hard instance [IW] and replace each terminal by a star of n terminals at zero distance Take hard instance [IW] and replace each terminal by a star of n terminals at zero distance The n terminals choose the same path to root The n terminals choose the same path to root cost of used edge becomes negligible cost of used edge becomes negligible Price of Anarchy: Lower Bound

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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. Round 1: players connect one by one to the root via best response. Solution T is reached after a sequence of arrivals t 1, t 2,…, t n. We show: Solution T is reached after a sequence of arrivals t 1, t 2,…, t n. We show: O(log 3 n ) c(T OPT ) O(log 3 n ) c(T OPT ) c(T) · O(log 2+ n ) c(T OPT ) c(T) · O(log 2+ n ) c(T OPT ) Round 2: players play in arbitrary order till NE is reached. c(T Nash ) Φ(T Nash ) Φ(T) Round 2: players play in arbitrary order till NE is reached. c(T Nash ) Φ(T Nash ) Φ(T)

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

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18 Charging -good Terminals t and t are -good terminals t and t are -good terminals t arrives first, then t arrives t arrives first, then t arrives Suppose there is a tree T spanning the -good terminals. Suppose there is a tree T spanning the -good terminals. Upon arrival t pays: cost(t) Upon arrival t pays: cost(t) Upon arrival, t pays at most: Upon arrival, t pays at most: cost(t) d + (1- ) ¢ cost(t) cost(t) d + (1- ) ¢ cost(t) r t t d cost(t)

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19 Charging -good Terminals (contd.) t 1,…, t k are -good terminals t 1,…, t k are -good terminals arrival order: t 1,…, t k arrival order: t 1,…, t k Charges decay at an exponential rate along a root – leaf path in T Upon arrival of t i it pays at most: Upon arrival of t i it pays at most: cost(t i ) · d i + (1- ) ¢ cost(t i-1 ) cost(t i ) · d i + (1- ) ¢ cost(t i-1 ) cost(t k ) · d k + d k-1 (1- ) cost(t k ) · d k + d k-1 (1- ) + d k-2 (1- ) 2 + … + d 1 (1- ) k-1 + d k-2 (1- ) 2 + … + d 1 (1- ) k-1 r t4t4 tktk dkdk t3t3 t1t1 t2t2 d1d1 d2d2 d3d3 d4d4 The charge to each edge e 2 T: · d(e) ¢ i (1- ) i ¢ n e (i)

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20 Auxiliary Tree T OPT is transformed to an auxiliary tree T defined on the set of -good terminals: The descendants of terminal t in T are terminals which have arrived after t. The descendants of terminal t in T are terminals which have arrived after t. c(T) · O(1/ ¢ logn) ¢ c(T OPT ) c(T) · O(1/ ¢ logn) ¢ c(T OPT ) The depth of T · O(1/ ¢ logn) The depth of T · O(1/ ¢ logn)

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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.

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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(log 4 n ) upper bound on PoA Setting = O(1/logn) O(log 4 n ) upper bound on PoA Setting cleverly O(log 3 n ) upper bound on PoA Setting cleverly O(log 3 n ) upper bound on PoA

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23 The Fractional Multicast Game Players split their connection to the source. Players split their connection to the source. A splittable multicast model. A splittable multicast model. r 12 3/4 1/4 3/4 1/4 x y Flow on (r, x): ¼ unit of flow is shared by users 1 & 2. ½ unit of flow is used only by user 1. Flow on (r, y): ¼ unit of flow is shared by users 1 & 2. ½ unit of flow is used only by user 2.

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24 The Fractional Multicast Game The cost of each flow fraction is split evenly between its users. The cost of each flow fraction is split evenly between its users. c e ·f e,n_e is the total cost of edge e. c e ·f e,n_e is the total cost of edge e. 1/43/47/8 f e,1 = 1/4 f e,2 = 3/4 f e,3 = 7/8 Total cost of user 3: c e · (1/12 + 1/4 + 1/8).

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

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

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27 Create a new graph G = (V, E) by replacing each edge e by n copies e 1, e 2, …, e n. Create a new graph G = (V, E) by replacing each edge e by n copies e 1, e 2, …, e n. Copy e j : should be used if j players use edge e. Copy e j : should be used if j players use edge e. The cost of a unit flow on copy j of edge e is c e / j. The cost of a unit flow on copy j of edge e is c e / j. The undirected graph is replaced by a directed flow network. The undirected graph is replaced by a directed flow network. Computing a Minimum Potential NE

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

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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. Heavily depends on the non-increasing property of the cost function. LP can be used: LP can be used: For any cost sharing mechanism that is cross-monotonic. For any cost sharing mechanism that is cross-monotonic. In case players are not restricted to have a common source. In case players are not restricted to have a common source. PoA of a minimum potential fractional NE is O(log n). PoA of a minimum potential fractional NE is O(log n).

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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. 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. 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. Building block: a variation of the Lund-Yannakakis hardness proof of approximating the set cover problem.

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31 The Weighted Multicast Game Each player i has a positive weight w i. Each player i has a positive weight w i. The cost share of each player is proportional to its weight. The cost share of each player is proportional to its weight. Integral: cost share of player i = c e · (w i / W e ) Integral: cost share of player i = c e · (w i / W e ) (W e – weight of players currently using e) (W e – weight of players currently using e) Fractional: weighted sharing on each fraction. 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].

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32 Thank You!

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