Communication Networks A Second Course Jean Walrand Department of EECS University of California at Berkeley

Non-Cooperative Games Comparing Nash Equilibria Three Problems in Networks Service Differentiation Multi-Provider Pricing Wi-Fi

Comparing Nash Equilibria Examples Pareto Minimax Risk-Dominant

Comparing Nash Equilibria Examples: LR T4, 41, 2 B2, 13, 3 LR T3, 21, 1 B 2, 3 LR T4, 42, 4 B5, 22, 3 LR T2, 2-100, 0 B0, -1001, 1

Comparing Nash Equilibria Pareto: LR T4, 41, 2 B2, 13, 3 Both players are strictly better off with the NE (T, L) than with the NE (B, R). They can safely assume that (T, L) will be played. (T, L) is the unique Pareto NE (the vector of rewards is not dominated component-wise by a different NE) Both players are strictly better off with the NE (T, L) than with the NE (B, R). They can safely assume that (T, L) will be played. (T, L) is the unique Pareto NE (the vector of rewards is not dominated component-wise by a different NE) A B

Comparing Nash Equilibria Minimax: Both (T, L) and (B, R) are Pareto: There is no other equilibrium where both players improve their rewards. What will happen? Minimax: (B, R) – Both A and B maximize their minimum reward. [Should be called maximin ….] Both (T, L) and (B, R) are Pareto: There is no other equilibrium where both players improve their rewards. What will happen? Minimax: (B, R) – Both A and B maximize their minimum reward. [Should be called maximin ….] A B LR T4, 31, 2 B2, 13, 4

Comparing Nash Equilibria Minimax: The Minimax is (B, L) However, this is not very credible …. If P1 thinks P2 chooses L, then P1 chooses T Also, if P2 thinks P1 chooses B, then P2 chooses R What will happen? The Minimax is (B, L) However, this is not very credible …. If P1 thinks P2 chooses L, then P1 chooses T Also, if P2 thinks P1 chooses B, then P2 chooses R What will happen? A B LR T4, 31, 1 B2, 23, 4

Comparing Nash Equilibria Minimax  Pareto Minimax: (T, L) Unique Pareto: (B, R) Would the players choose (B, R)? Minimax: (T, L) Unique Pareto: (B, R) Would the players choose (B, R)? LR T2, 22, 1 B1, 25, 5

Comparing Nash Equilibria Risk-Dominant (T, L) is the unique Pareto NE However, by playing L, P2 faces a big risk if P1 plays B P2 reduces his risk by playing R Knowing this, P1 may prefer to play B P1 faces a smaller risk by playing B than P2 does by playing L … This suggests that (B, R) is a less risky NE. (T, L) is the unique Pareto NE However, by playing L, P2 faces a big risk if P1 plays B P2 reduces his risk by playing R Knowing this, P1 may prefer to play B P1 faces a smaller risk by playing B than P2 does by playing L … This suggests that (B, R) is a less risky NE. LR T2, 2-5, 0 B0, -51, 1

Comparing Nash Equilibria Risk-Dominant Concept: Risk-Dominance Define p as follows: If P(P2=L) ≥ p, then P1 = T is preferable.  If P(P2=R) ≥ 1 - p, then P1 = B is preferable If p > 0.5, we say that (B, R) is risk-dominant; else (T, L) is risk-dominant In this case, p = 0.75  (B, R) is risk-dominant Concept: Risk-Dominance Define p as follows: If P(P2=L) ≥ p, then P1 = T is preferable.  If P(P2=R) ≥ 1 - p, then P1 = B is preferable If p > 0.5, we say that (B, R) is risk-dominant; else (T, L) is risk-dominant In this case, p = 0.75  (B, R) is risk-dominant LR T2, 2-5, 0 B0, -51, 1

Comparing Nash Equilibria Risk-Dominant Let’s do the math … If P(P2=L) ≥ p, then P1 = T is preferable U 1 (T) ≥ 2p – 5(1 – p); U 1 (B) ≤ 0p + 1(1 – p) Minimum value of p is s.t. 1 – p = 2p – 5(1 – p)  p = 0.75 Let’s do the math … If P(P2=L) ≥ p, then P1 = T is preferable U 1 (T) ≥ 2p – 5(1 – p); U 1 (B) ≤ 0p + 1(1 – p) Minimum value of p is s.t. 1 – p = 2p – 5(1 – p)  p = 0.75 LR T2, 2-5, 0 B0, -51, 1

Comparing Nash Equilibria Risk-Dominant AB A  2, 2-0.5, 0 B0, -0.51, 1 LR T2, 2-5, 0 B0, -5  1, 1

Comparing Nash Equilibria Risk-Dominant AB A9, 90, 8 B8, 07, 7 0 1¾ t BB AA N(1) Define: G(t) = Game where P2 randomizes his strategy with probability 1 – t N(t) = set of NE for G(t) If there is a continuous graph {t, f(t) in N(t)}, then f(1) defines a Linear-Tracing Risk-Dominant NE Related Definition:  Linear-Tracing

Three Problems 1.Service Differentiation 2.Multi-Provider Network 3.Pricing Wi-Fi 4.Conclusions

Three Problems 1.Service Differentiation Market segmentation Capture willingness to pay more for better services

Three Problems (cont.) 2.Multi-Provider Network Incentives for better services through all providers  Improved Services & Revenues

Three Problems (cont.) 3.Wi-Fi Access Incentives to open private Wi-Fi access points  Ubiquitous Access

Three Problems (cont.) Note: Related problems P2P Incentives Incentives for Security Fair sharing among content and network providers

June 28, 2002Preliminary Results19 Service Differentiation Model Examples Proposal Joint work with Linhai He

June 28, 2002Preliminary Results20 Service Differentiation Model Model Examples Proposal Joint work with Linhai He

June 28, 2002Preliminary Results21 Model Two possible outcomes: 1.Users occupy different queues (delays = T 1 & T 2 ) 2.Users share the same queue (delay = T 0 ) If users do not randomize their choices, which outcome will happen? Each user chooses the service class i that maximizes his/her net benefit p1p1 p2p2 Users A B H L Priority

June 28, 2002Preliminary Results22 Model (cont) p1p1 p2p2 A B H L HL B H L A f 1 (T 0 ) – p 2 f 1 (T 1 ) – p 1 f 1 (T 2 ) – p 2 f 1 (T 0 ) – p 1 A’s benefit T 1 < T 0 < T 2 f i (.) nonincreasing TOC TOC – Service Differentiation  ModelService Differentiation B’s benefit f 2 (T 0 ) – p 1 f 2 (T 2 ) – p 2 f 2 (T 1 ) – p 1 f 2 (T 0 ) – p 2

June 28, 2002Preliminary Results23 Service Differentiation Model Examples Examples Proposal Joint work with Linhai He

June 28, 2002Preliminary Results24 Example 1 HL H L B A 9 – 4 = 5 9 – 1 = 8 14 – 4 = 10 5 – 1 = 4 14 – 4 = 10 p1p1 p2p2 A B H L f(T 1 ) = 14 f(T 0 ) = 9 f(T 2 ) = 5 p 1 = 4 p 2 = 1 Here, f i (.) = f(.)

June 28, 2002Preliminary Results25 Example 1 HL H L B A NE Assume A picks H. Should B choose H or L? Assume A picks H. Should B choose H or L? Assume A picks H. B should choose H. Assume A picks H. B should choose H. Assume A picks L. Should B choose H or L? Assume A picks L. Should B choose H or L? Assume A picks L. B should choose H. Assume A picks L. B should choose H. B  H. Since B chooses H, A should also choose H. NE = Nash Equilibrium

June 28, 2002Preliminary Results26 Example 1 HL H L B A NE A and B choose H, get rewards equal to 5. If they had both chosen L, their rewards would have been 8! A and B choose H, get rewards equal to 5. If they had both chosen L, their rewards would have been 8! Prisoner’s Dilemma!

June 28, 2002Preliminary Results27 Example 2 HL H L B A 9 – – – – p1p1 p2p2 A B H L T 1 : 13, 11 T 0 : 9, 9 T 2 : 7, 5 p 1 = 4 p 2 = 1 No Pure Equilibrium f 0 f 1

June 28, 2002Preliminary Results28 Example 3 Extension to many users Equilibrium exists if 9  0 s.t. willingness to pay total load in class i (Indeed, ) Also, the other users prefer L. Note: T 1 and T 2 depend on the split of customers. In this equilibrium, users with prefer H.

June 28, 2002Preliminary Results29 Example 3 Analysis of equilibria: inefficient equilibrium unstable equilibrium Here, f is a concave function and strict-priority scheduling is used.  p1-p2p1-p2  {f(T 1 )-f(T 2 )}

June 28, 2002Preliminary Results30 Service Differentiation Model Examples Proposal Proposal Joint work with Linhai He

June 28, 2002Preliminary Results31 Proposal Dynamic Pricing Fixed delay + dynamic price Provider chooses target delays for both classes Adjust prices based on demand to guarantee the delays Users still choose the class which maximizes their net benefit

June 28, 2002Preliminary Results32 Multiprovider Network Model Nash Game Revenue Sharing Joint work with Linhai He

June 28, 2002Preliminary Results33 Multiprovider Network Model Model Nash Game Revenue Sharing Joint work with Linhai He

Model + p 1 + p 2 p1+ p2p1+ p2 Monitor marks and processes inter- network billing info Pricing per packet

June 28, 2002Preliminary Results35 Multiprovider Network Model Nash Game Nash Game Revenue Sharing Joint work with Linhai He

Nash Game: Formulation 12 p1p1 p2p2 D Demand = d(p 1 +p 2 ) C1C1 C2C2 A game between two providers Different solution concepts may apply, depend on actual implementation Nash game mostly suited for large networks Provider 1 Provider 2

Nash Game: Result 1.Bottleneck providers get larger share of revenues than others 2.Bottleneck providers may not have incentive to upgrade 3.Efficiency decreases quickly as network size gets larger (revenues/provider drop with size)

June 28, 2002Preliminary Results38 Multiprovider Network Model Nash Game Revenue Sharing Revenue Sharing Joint work with Linhai He

Revenue Sharing Improving the game Model Optimal Prices Example

Revenue Sharing Improving the game Improving the game Model Optimal Prices Example

Revenue Sharing - Improving the Game Possible Alternatives Centralized allocation Cooperative games Mechanism design Our approach: design a protocol which overcomes drawbacks of non-cooperative pricing is in providers’ best interest to follow is suitable for scalable implementation

Revenue Sharing Improving the game Model Model Optimal Prices Example

Revenue Sharing - Model Providers agree to share the revenue equally, but still choose their prices independently 1 2 p1p1 p2p2 D Demand = d(p 1 +p 2 ) C1C1 C2C2 Provider 1 Provider 2

Revenue Sharing Improving the game Model Optimal Prices Optimal Prices Example

Revenue Sharing - Optimal Prices # of providers Lagrange multiplier on link i “locally optimal” total price for the route sum of prices charged by other providers  A system of equations on prices

Revenue Sharing - Optimal Prices (cont.) For any feasible set of  i, there is a unique solution: On the link i with the largest ,  * ), p i * = N  * + g( p i * ) On all other links, p j * = 0  Only the most congested link on a route sets its total price

Revenue Sharing - Optimal Prices (cont.) {i}{i} {pi*}{pi*} {dr*}{dr*}  a Nash game with  i as the strategy It can be shown that a Nash equilibrium exists in this game. Each provider solves its  i based on local constraints

Revenue Sharing - Optimal Prices (cont.) Comparison with social welfare maximization (TCP) Social: Sharing: Incentive to upgrade Upgrade will always increase bottleneck providers’ revenue  A tradeoff between efficiency and fairness

Revenue Sharing - Optimal Prices (cont.) Efficient when capacities are adequate It is the same as that in centralized allocation Revenue per provider strictly dominates that in Nash game

Revenue Sharing - Optimal Prices (cont.) A local algorithm for computing  i that can be shown to converge to Nash equilibrium:

Revenue Sharing - Optimal Prices (cont.) 1 i d hop count N r =0 congestion price  r =0 flows on route r N r =N r +1  r = max(  r,  i ) A possible scheme for distributed implementation … … … No state info needs to be kept by transit providers.

Revenue Sharing Improving the game Model Optimal Prices Example Example

C 1 =2 C 2 =5 C 3 =3 demand = 10 exp(-p 2 ) on all routes r1r1 r2r2 r3r3 r4r4 ii link 1 link 3 link 2 prices p2p2 p3p3 p1p1 p4p4

Wi-Fi Pricing Motivation Web-Browsing File Transfer Joint work with John Musacchio

Wi-Fi Pricing Motivation Motivation Web-Browsing File Transfer Joint work with John Musacchio

Motivation Path to Universal WiFi Access Massive Deployment of base stations for private LANs Payment scheme might incentivize base station owners to allow public access. Direct Payments Avoid third party involvement. Transactions need to be “self enforcing” Payments: Pay as you go: In time slot n, - Base Station proposes price p n - Client either accepts or walks away What are good strategies?

Wi-Fi Pricing Motivation Web-Browsing Web-Browsing File Transfer Joint work with John Musacchio

Web Browsing Client Utility U = Utility per unit time K = Intended duration of connection Random variable in [0, 1] Known to client, not to BS Random variable in {1, 2, …} Known to client, not to BS BS Utility p 1 + p 2 + … + p N U.min{K, N} N = duration

Web Browsing Theorem Perfect Bayesian Equilibrium: Client accepts to pay p as long as p ≤ U BS chooses p n = p* = arg max p p P(U ≥ p) Note: Surprising because BS learns about U …

Wi-Fi Pricing Motivation Web-Browsing File Transfer File Transfer Joint work with John Musacchio

File Transfer Client Utility K.1{K ≤ N} BS Utility p 1 + p 2 + … + p N K = Intended duration of connection Random variable in {1, 2, …} Known to client, not to BS N = duration

File Transfer Theorem Perfect Bayesian Equilibrium: Client accepts to pay 0 at time n < K p ≤ K at time n = K BS chooses a one-time-only payment pay n* at time n* = arg max n nP(K = n) Note: True for bounded K. Proof by backward induction. Unfortunate ….

Three Problems: Conclusions Dynamic Pricing to adjust QoS Cooperative pricing -> distributed algorithm Web browsing -> constant price File transfer -> one-time price References: Linhai He and Jean Walrand, “Pricing and Revenue Sharing for Internet Service Providers.”. To appear in JSAC 2006.Pricing and Revenue Sharing for Internet Service Providers John Musacchio and Jean Walrand, “Game Theoretic Modeling of WiFi Pricing,” Allerton, 10/1/2003Game Theoretic Modeling of WiFi Pricing References: Linhai He and Jean Walrand, “Pricing and Revenue Sharing for Internet Service Providers.”. To appear in JSAC 2006.Pricing and Revenue Sharing for Internet Service Providers John Musacchio and Jean Walrand, “Game Theoretic Modeling of WiFi Pricing,” Allerton, 10/1/2003Game Theoretic Modeling of WiFi Pricing