1. Reference Cole, Dodis, Roughgarden (2006) How much can taxes help selfish routing? Journal of Computer and System Sciences Kearns, Littman, Singh (2001) Graphical Models for Game Theory, UAI Monderer (2007) Multipotential Games, IJCAI Nash (1950) Equilibrium Points in N-person Games Odlyzko (1997) A Modest Proposal for Preventing Internet Congestion Ros, Tuffin (2004) A Mathematical Model of the Paris Metro Pricing Scheme, Computer Networks Paris Metro PricingExpedited ServiceBraess’ ParadoxGeneral Networks Game-Theoretic Analysis of Network Quality-of-Service Pricing David R.M. ThompsonAlbert Xin JiangKevin Leyton-Brown daveth@cs.ubc.cajiang@cs.ubc.cakevinlb@cs.ubc.ca Introduction Network System Q: Does a network provide good quality of service? A: That depends on what its users want from it. Game Solver Normally impractical: Nash equilibria are too expensive to compute (O( 2 2 n ) where n is number of users) Action Graph Games exploit structure for massive speed gain: [Bhat & Leyton-Brown, 2004; Jiang & Leyton-Brown, 2006] Anonymity: other users’ behavior affects my QoS, not their identities Context specific independence: my QoS is unaffected by traffic on links I’m not using Can be treated as a black-box Input: network Output: usage pattern Game Theoretic Model Different users have different values for quality of service: User’s experienced QoS (e.g. latency) influenced by other users’ actions (which cause congestion) This interdependence means game theory applies. Definition: “Nash equilibrium”: a stable state where no user wants to change their action, given the actions of everyone else [Nash, 1950] Implications and Conclusions The equilibrium of an AGG would allow us to answer questions about the proposed network: What paths through the network would the users choose? How much load would occur on each link? What is each user’s utility? (i.e. how happy are they with the network?) Definition: “Social welfare”: sum of all parties’ utilities (users and network providers) Definition: “Economic efficiency”: maximizing social welfare TCP/IP Back-off Network System 2 TCP/IP users, 1 shared link Converges to Equal division of bandwidth Limited congestion Game Theoretic Model Suppose user 1 hacks his TCP/IP back-off implementation: Converges to Unequal share of bandwidth More congestion Suppose both users hack: Converges to Equal share of bandwidth Even more congestion Game Solver Equivalent to “prisoner’s dilemma” Only equilibrium is for both users to hack User 2 NormalHacked User 1 Normal-1,-1-4,0 Hacked0,-4-3,-3 Implications and Conclusions The only equilibrium is the least economically efficient state. Fortunately, TCP/IP hacks have a cost to adopt and hackers have a disincentive to share their work. Introduction First class Economy class Q: Why charge different prices for identical service? A: Because they’re expensive, first-class cars are less crowded. Same concept applied to highway traffic: Toronto 407’s toll is tuned to control congestion Network System Q: Can we use this idea to prevent internet congestion? [Odlyzko, 1997; Ros & Tuffin, 2004] Linear, additive model of latency: Delay = ∑( Usage ) / Bandwidth A “perfect” fair queue of unlimited length Game Theoretic Model 18 low priority users, 2 high priority users Linear model of utility: Utility = –Delay × ValueForTime – LinkToll Utility measured in $ (cost-benefit trade-off of QoS) Game Solver Iterate over a range of prices: $0.00 to $2.00 in $0.01 increments AGG solver finds usage pattern given costs Implications and Conclusions Economically efficient between $0.72 and $1.10 (Cost of latency minimized) Most profit goes to network provider Significant waste: Costly link sits idle while users wait in free link’s queue Network System Q: How does a tiered QoS system compare with Paris Metro pricing? Consider the same network and assumptions as in Paris Metro pricing example Add “Perfect” expedited service: Expedited traffic unaffected by non-expedited Game Theoretic Model Same utility and user model as Paris Metro pricing example Game Solver Iterate over a range of prices: $0.00 to $2.00 in $0.01 increments AGG solver finds usage pattern given costs: Implications and Conclusions Economically efficient when cost > $0.72 (Cost of latency minimized) Most profit goes to users. No waste: Load always uniformly balanced Network System Future extensions to network model: Arbitrary network topology Richer models of usage (e.g. bandwidth consumption, burstiness) Richer models of tiered service (i.e. imperfectly expedited service) Network System Delay of a path is the sum of delays of link segments along the path Game Theoretic Model 20 users Each can choose any path from s to t At equilibrium: flow split between 2 paths Adding a link At equilibrium: all users choose path s,u,v,t All users are worse off Pricing Put price on link (u,v) When users have same values: (u,v) useless Q: What happens if users have different values? Game Solver AGG solver finds usage pattern given costs: Implications Economically efficient between $0.81 and $9.50 Most profit goes to users More efficient than without the link (u,v) Related Work Network Model: Ros & Tuffin (2004): game-theoretic analysis of Paris-Metro Pricing Cole et al (2006): analyzes putting taxes on links to reduce congestion (neither paper modeled users with different values for latency) Game Representations: Kearns et al (2001): Graphical Games exploits strict independence structure cannot compactly represent games here Monderer (2006): Player-specific congestion games Can compactly represent games here Did not focus on computation of Nash equilibria Game Theoretic Model Future extensions to user model: Arbitrary source and destination nodes Uncertainty about the types of other agents (i.e. Bayesian games) Game Solver All proposed model extensions are possible within existing AGG framework When utility, latency functions have simple structure (e.g. path latency = sum of link latencies, path bandwidth = min of link bandwidths) even more optimization may be possible Game Generator Object-Oriented Python API: Takes Network object as input Generates AGG file Launches AGG solver Interprets results Restricted to parallel paths or Braess-structured networks, with perfect expedited service Arbitrary latency functions for each link, L: f L (# of users) → Real value Supports richer QoS measure than just latency: f L (# of users) → Q where Q is an arbitrary set (e.g. vectors of features such as bandwidth, latency, probability of packet loss) Arbitrary utility functions: f U (Q) → Real value