1 Algorithmic Game Theoretic Perspectives in Networking Dr. Liane Lewin-Eytan

2 Prisoner’s Dilemma Scenario: Two suspects in a crime are put into separate cells. If they both confess, each will be sentenced into 3 years in prison. If only one of them confesses, he will be freed and used as a witness against the other, who will receive a sentence of 4 years. If neither confesses, they will both be convicted of a minor offense and spend one year in prison. 4,01,1 3,30,4 confess don ’ t confess confess 1 equilibrium: ( confess, confess )

3 Games in Networks Users with a multitude of diverse economic interests sharing a Network (Internet) browsers browsers routers routers servers serversSelfishness: Parties deviate from their protocol if it is in their interest

4 Motivation 1. Networking Traditional networks – single entity with single control objective. Traditional networks – single entity with single control objective. Modern networking – interaction of many entities controlled by different parties. Modern networking – interaction of many entities controlled by different parties.

5 Motivation 2. Games Users act according to their individual interests so as to maximize their own objective functions. Users act according to their individual interests so as to maximize their own objective functions. A user makes selfish decisions based on the state of the network, which depends on the behavior of the other users. A user makes selfish decisions based on the state of the network, which depends on the behavior of the other users.  non-cooperative network games.

6 Motivation 3. Algorithms Computational and algorithmic issues arise in large and complex games motivated by large decentralized computer networks (the Internet). Computational and algorithmic issues arise in large and complex games motivated by large decentralized computer networks (the Internet).

7 NetworkingGames Algorithms Non-Cooperative Network Games Algorithmic Game Theory Algorithmic Perspectives of Game Theory in (Large Scale) Networks

8 A Simple Game: Load Balancing Each job wants to be on a lightly loaded machine machine 1 machine 2 With coordination we can arrange them to minimize load Example: load of 4

9 A Simple Game: Load Balancing Each job wants to be on a lightly loaded machine Without coordination? Stable arrangement: No job has incentive to switch Example: some have load of 5

10 Games: Setup A set of players (in example: jobs) A set of players (in example: jobs) for each player, a set of strategies (which machine to choose) for each player, a set of strategies (which machine to choose) Game: each player picks a strategy For each strategy profile (a strategy for each player)  a payoff to each player (load on selected machine)

11 Nash Equilibrium A set of actions (strategy choices), one per player, where no player can unilaterally improve its performance by changing its strategy. The Nash equilibrium solutions of a game are its stable operating points (stable strategy profile). The Nash equilibrium solutions of a game are its stable operating points (stable strategy profile).

12 Quality of Outcome: Goal of the Game Personal objective for player i: min load L i Overall objective? Social Welfare:  i L i Makespan: max i L i

13 Routing network: ℓ e (x) = x s t Delay as a function of load: x unit of load  causes delay ℓ e (x) Load Balancing and Routing Load balancing: jobs machines ℓ e (x) = x Allow more complex networks st x1 x 1 0

14 Framework A network shared by selfish users. A network shared by selfish users. Each resource has a cost to be paid by its users. Each resource has a cost to be paid by its users. Performance of a user = its total payment = sum of payments for all the resources it uses. Performance of a user = its total payment = sum of payments for all the resources it uses. Two fundamental models: Two fundamental models: The congestion model. The congestion model. The cost sharing model. The cost sharing model.

15 Resource cost is fixed. Resource cost is fixed. 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. Each user has a favorable effect on the performance of other users. Resource cost: Resource cost: Modeled by a load dependent function. Modeled by a load dependent function. Non-decreasing in the load of the resource. Non-decreasing in the load of the resource. Each user has a negative effect on the performance of other users. Each user has a negative effect on the performance of other users. Cost Sharing Model Congestion Model

16 The Game Perspective Strategy space of each player: subsets of resources. Strategy space of each player: subsets of resources. Cost allocation method defines the rules of the game:  determines the mutual influence among the players. Cost allocation method defines the rules of the game:  determines the mutual influence among the players. Each player knows the rules of the underlying game. Each player knows the rules of the underlying game. Players are rational: a player chooses a strategy that minimizes its total payment. Players are rational: a player chooses a strategy that minimizes its total payment.

17 The Congestion Model

18 Routing with Delay Edge-delay is a function ℓ e () of the load on the edge e Assume ℓ e (x) continuous and non-decreasing in load x on edge e. Assume ℓ e (x) continuous and non-decreasing in load x on edge e. st x 1 x 1 0

19 One unit of flow sent from s to t One unit of flow sent from s to t A stable solution: Users control an infinitesimally small amount of flow. x st 1 Flow =.5 Traffic on upper edge is envious. Example on two links No-one is better off x st 1 Flow = 1 Flow = 0

20 Model of Routing Game A directed graph G = (V,E) A directed graph G = (V,E) source–sink pairs s i,t i for i=1,..,k source–sink pairs s i,t i for i=1,..,k rate r i  0 of traffic between s i and t i for each i=1,..,k rate r i  0 of traffic between s i and t i for each i=1,..,k r 1 =1 st x1.5 x 1 Load-balancing jobs wanted min load Here want minimum delay: delay adds along path

21 Goal of the Game Personal objective: choose a path minimizing ℓ P (f) = sum of latencies of edges along path P Overall objective: minimize C(f) = total latency of a flow f =  e f e ℓ e (f) =social welfare

22 Network Routing Game Flow represents cars on highways cars on highways packets on the Internet packets on the Internet User goal: Find a path minimizing delay  true for cars, packets?: users do not choose paths on the Internet: routers do! With delay as primary metric  router protocols choose shortest path!

23 Braess’s Paradox Original Network Cost of Nash flow =2(1.5*0.5)=1.5 st x1.5 x 1 Added edge: Effect? st x1 x 1.5 0

24 Braess’s Paradox Original Network Added edge: Cost of Nash flow = 2 All the flow has increased delay! st x1 x Cost of Nash flow = 2(1.5*0.5)=1.5 st x1.5 x 1

25 Some Results Theorem (Roughgarden-Tardos’00) In a network with linear latency functions In a network with linear latency functions i.e., of the form ℓ e (x)=a e x+b e i.e., of the form ℓ e (x)=a e x+b e the cost of a Nash flow is at most 4/3 times that of the minimum-latency flow the cost of a Nash flow is at most 4/3 times that of the minimum-latency flow  Price of Anarchy = 3/4

26 Some Results Theorem 1 (Roughgarden-Tardos’00) In a network with linear latency functions In a network with linear latency functions i.e., of the form ℓ e (x)=a e x+b e i.e., of the form ℓ e (x)=a e x+b e the cost of a Nash flow is at most 4/3 times that of the minimum-latency flow the cost of a Nash flow is at most 4/3 times that of the minimum-latency flow st x 1 r=1 x 1 0 x st 1 Flow =.5 Nash cost 1 optimum 3/4Nash cost 2 optimum 1.5

27 The Price of Anarchy Typically, Nash equilibrium outcomes do not optimize the overall network performance. Typically, Nash equilibrium outcomes do not 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.

28 The Cost Sharing Model

29 Multicast A source simultaneouslytransmits the same data to a group of destinations. A source simultaneously transmits the same data to a group of destinations. Messages are transmitted over each link of the network only once. Messages are transmitted over each link of the network only once. Multicast nodes create copies when the links to the destinations split. Multicast nodes create copies when the links to the destinations split. Multicast routing increases network efficiency. Multicast routing increases network efficiency. r t1t1 t2t2 t4t4 t5t5 t6t6 t3t3

30 A Cost Sharing Multicast Game A special source node (root) r, and a set N of n receivers (players). A special source node (root) r, and a set N of n receivers (players). A player’s strategy is a routing decision – the choice of a route from its terminal to r. A player’s strategy is a routing decision – the choice of a route from its terminal to r. Egalitarian cost sharing mechanism: the cost of each edge is evenly split among its downstream receivers. c e i (s) = c e / n e (s) Egalitarian cost sharing mechanism: the cost of each edge is evenly split among its downstream receivers. c e i (s) = c e / n e (s)

31 Egalitarian Cost Sharing Mechanism Payment of t 1 : c 1 /4 Payment of t 2 : c 1 /4 + c 2 Payment of t 3 : c 1 /4 + c 3 /2 Payment of t 4 : c 1 /4 + c 3 /2 + c 4 r t1t1 t2t2 t3t3 t5t5 t6t6 t4t4 c1c1 c2c2 c3c3 c4c4

32 Goal of the Game Personal objective: choose a path to the root minimizing payment. Overall objective: minimize C(T) = total cost of T =  e  T c e = social welfare = Steiner tree !

33 Potential Function Multicast game admits a potential function. Multicast game admits a potential function. Potential function Φ of a solution T [Rosenthal `73] : Potential function Φ of a solution T [Rosenthal `73] : Exact potential: Exact potential: Change in potential = change in payoff of player making a move Global / Local optimum of Φ corresponds to a NE. Global / Local optimum of Φ corresponds to a NE.

34  Price of anarchy can be as bad as  (n). 1 st n OPT (= Best NE) all players use cheap edge each pays 1/n total cost = 1 1 st n Worst NE all players use expensive edge each pays n/n=1 total cost = n Price of Anarchy

35 The Price of Stability Price of anarchy: Price of anarchy: Can be unbounded. Can be unbounded. Also captures “non-interesting” equilibria. Also captures “non-interesting” equilibria. Price of Stability: The ratio between the cost of the best Nash solution and the cost of OPT. Outcome of scenarios in the ‘middle ground’ between centrally enforced solutions and selfish behavior. Outcome of scenarios in the ‘middle ground’ between centrally enforced solutions and selfish behavior. E.g.: central entity can enforce the initial operating point. E.g.: central entity can enforce the initial operating point.

36 Price of stability – upper bound is O(log n). Price of stability – upper bound is O(log n). c(T Nash )  Φ(T Nash )  Φ(T initial )  log n ∙c(T initial ) proof: edge cost c e with n e > 0 users edge cost c e with n e > 0 users edge potential with n e > 0 users edge potential with n e > 0 users  e =c e ·(1+1/2+1/3+…+1/n e )  Ratio at most H n =O(log n)  Ratio at most H n =O(log n) Price of Stability

37 Example: Bound is Tight 1 1 n n t ... n-1 0 1

38 Example: Bound is Tight 1 1 n n t ... n cost(OPT) = 1+ε

39 Example: Bound is Tight 1 1 n n t ... n cost(OPT) = 1+ε …but not a NE: player n pays (1+ε)/n, could pay 1/n

40 Example: Bound is Tight 1 1 n n t ... n so player n would deviate

41 Example: Bound is Tight 1 1 n n t ... n now player n-1 pays (1+ε)/(n-1), could pay 1/(n-1)

42 Example: Bound is Tight 1 1 n n t ... n so player n-1 deviates too

43 Example: Bound is Tight 1 1 n n t ... n Continuing this process, all players defect. This is a NE! (the only Nash) cost = … + Price of Stability is H n = Θ(log n) ! 1 2 n

44 Best response dynamics : each player, in its turn, selects a strategy minimizing its cost (or maximizing its profit). Best response dynamics : each player, in its turn, selects a strategy minimizing its cost (or maximizing its profit). Natural game course continues until a NE is reached. Natural game course continues until a NE is reached. PoA may depend on the initial game configuration. PoA may depend on the initial game configuration. A natural starting point: empty configuration. A natural starting point: empty configuration. Best Response Dynamics

45 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, starting from an empty configuration? n-2n-1 NE OPT

46 Some Results (Chuzhoy et al. ‘06) Upper bound of on the PoA of best-response dynamics in case players join the game sequentially starting from an ‘empty’ configuration.  was improved to O(log 3 n) by Charikar et al. Upper bound of on the PoA of best-response dynamics in case players join the game sequentially starting from an ‘empty’ configuration.  was improved to O(log 3 n) by Charikar et al. Lower bound of on the PoA of this game. Lower bound of on the PoA of this game. Computing a NE minimizing Rosenthal’s potential function is NP-hard. Computing a NE minimizing Rosenthal’s potential function is NP-hard.

47 Thank You Thank You