Games in Network Routing
Nikhil Shetty
EE228A
Overview
• Basics
• Costs, social optimum
• How Bad is Selfish Routing? - Tardos, Roughgarden
• Mechanisms to deal with it
• Algorithmic Mechanism Design
• E.g. 1: BGP Routing
• E.g. 2: Ad Hoc Routing
• 2.1 -- SPRITE
• 2.2 – Ad-hoc VCG
Basics
• Network of nodes.
• Latency on link function of traffic.
• Users selfish not malicious.
• Each user wants to minimize latency for his traffic.
• Results in Nash Equilibrium between flows.
Basics (contd)
• Cost is the product of latency and traffic.
• Social optimum – minimize sum of costs of all the flows.
• Nash Equilibrium may not be socially optimal.
Example
• One unit of flow from s to t.
Example (contd)
• Nash Equilibrium – envy-free solution
• No part of the flow is better off.
Braess Paradox
• Nash Flow is optimal
Braess Paradox (contd)
• Nash Flow performs worse.
Another example
Cost of selfish routing
• Also termed Price of Anarchy
• Characterized by the ratio of
• Cost of selfish equilibrium to
• Cost of optimal solution
• Users are selfish
• Minimize personal latency
• Don’t care for the social optimal solution
Model (Tardos-Roughgarden)
• Graph G = (V,E)
• k source-destination pairs {s1, t1},{s2, t2},….
• Rate ri traffic between {s_i, t_i}
• fe sum of flows on edge e.
Nash Equilibrium for flows
Optimal Solution
Optimal Solution
• Solution to the convex optimization.
• Minimize the sum of costs on all links in the network.
• Local optimum – marginal cost of moving from one path to another is same.
• Flow is globally optimal if latency functions are convex.
Beckman et al.
• Compare the two inequalities.
• We can relate a Nash equilibrium to the social optimal by replacing the latency functions on the links.
Another result
Proof
Worst case ratio
• Linear latency functions
• Use the comparison of Nash and optimum
Few more lemmas
And finally…
In short
Algorithmic Mechanism Design
• AMD – between CS and Economics.
• Design mechanisms with polynomial time complexities.
• Major contribution by Nisan and Ronen.
Algorithmic Mechanism Design
mechanism
t1
t2
tn
a_1
a_2
a_n p_n
p_1
p_2
O
AMD
• Agent’s utility or valuation
• Strategy-proof mechanism
• Truth-telling is the dominant strategy
Example: BGP routing
• Network of AS – each AS is a node.
• AS may carry transit traffic. Increases congestion in its network. Will be a node on the path.
• Assume graph is biconnected.
• C is the vector of transit costs.
• Find least-cost path between AS’s.
BGP
• Cost incurred by transit node k. T is the amount of traffic. I is an indicator function for node on a path.
• Total cost to society to be minimized.
BGP
• But all this can be done only if we know the true costs.
• That has to be ensured using algorithmic mechanism design.
• Utility of the players.
• By AMD,
• VCG Mechanism
Ad hoc-VCG
• Route discovery and cost calculation.
• Nodes forward the minimum energy required for transmission and cost of the energy.
• Destination chooses the lowest-cost path.
• Destination calculates payment based on VCG.
Ad hoc-VCG
Ad hoc-VCG properties
Ad hoc-VCG properties
• Overpayment is limited.
Some ideas
• Ad hoc networks still haven’t taken off commercially.
• Reputation systems good for distributed computation – bad in actual cost quantification.
• SPRITE/Ad hoc-VCG good properties of truthfulness. – not distributed (may require central bank).
• Must focus on infrastructure operator-facilitated ad hoc networks (my guess) to make it “juicy”for them to take up.
Some more ideas…
• Use a mixture of reputation and centralized.
• Operators might have additional issues like revenue maximization and interference control.
• Could be interesting issues to investigate.
References
• T. Roughgarden, ´E. Tardos; How Bad is Selfish Routing in: Journal ofthe ACM, 49(2), pp. 236-259, March 2002.
• J. Feigenbaum, Ch. Papadimitriou, R. Sami,
S. Shenker; A BGP-based Mechanism for Lowest-Cost Routing; in: Proceedings oft he 21st ACM Symposium on Principles of Distributed Computing (PODC), pp.173-182, July 2002.
References
• S. Zhong, Yang Richard Yang, J. Chen; Sprite: A Simple, Cheat-Proof, Credit-Based System for Mobile Ad-hoc Networks; in: Proceedings ofINFOCOM 2003,pp. 1987-1997, March 2003.
• L. Anderegg and S. Eidenbenz; Ad hoc-VCG: a truthful and cost-efficient routing protocol for mobile ad-hoc networks with selfish agents.