Games in Network Routing - people.eecs.berkeley.eduwlr/228S06/RoutingGames.pdf · Example: BGP routing • Network of AS – each AS is a node. • AS may carry transit traffic. Increases

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.