Communication Networks A Second Course Jean Walrand Department

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

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

Yang-Hajek: VCG-Kelly Mechanism • • • Motivation Problem Mechanism Kelly’s Mechanism Problems with strategic

Yang-Hajek: VCG-Kelly Mechanism • • • Motivation Problem Mechanism Kelly’s Mechanism Problems with strategic players VCG Mechanism Hajek-Yang’s combination: VCG-Kelly Efficiency Stability Yank and Hajek: VCG-Kelly Mechanisms for Allocation of Divisible Goods: Adapting VCG Mechanisms to One-Dimensional Signals. 40 th Annual Conference on Information Sciences and Systems (CISS 06), Princeton, NJ, Mar 22 -24, 2006. .

Motivation • Problem: Efficient sharing of bandwidth in networks • Price-Taking Users: Kelly (One-dimensional

Motivation • Problem: Efficient sharing of bandwidth in networks • Price-Taking Users: Kelly (One-dimensional bids) • Strategic Users: VCG (Infinite-dimensional bids: utility functions) • Combination: Efficient mechanism with onedimensional bids for strategic users

Problem (continued) Necessary and sufficient conditions

Problem (continued) Necessary and sufficient conditions

Mechanism “bids”

Mechanism “bids”

Kelly Mechanism Network problem Solution: User problem

Kelly Mechanism Network problem Solution: User problem

Kelly Mechanism (continued) fi(x) = log(xi) surrogate valuation: Wi(x) = wilog(xi) one-dimensional signals: wi

Kelly Mechanism (continued) fi(x) = log(xi) surrogate valuation: Wi(x) = wilog(xi) one-dimensional signals: wi instead of Ui(x)

Problems with strategic players 1) NE may not exist for Kelly’s mechanism with strategic

Problems with strategic players 1) NE may not exist for Kelly’s mechanism with strategic players: Necessary conditions for optimality imply x = (4, 1, 3), which is not an optimal point.

Problems with strategic players 1. 2) NE exists, it may be very inefficient: Social:

Problems with strategic players 1. 2) NE exists, it may be very inefficient: Social: x 0 = 0, xi = 1, I = 1, …, L NE: x 0 = g(g + L)-1, xi = (g + 1)-1 Loss: (g 2 + L)[L(g + 1)]-1 Let g 2 = L and let L

VCG Mechanism Incentive-Compatible: w() = u(. ) is dominant!

VCG Mechanism Incentive-Compatible: w() = u(. ) is dominant!

VCG Mechanism (continued) The bids are functions difficult to implement!

VCG Mechanism (continued) The bids are functions difficult to implement!

VCG-Kelly Mechanism

VCG-Kelly Mechanism

VCG-Kelly Mechanism (continued)

VCG-Kelly Mechanism (continued)

Efficiency Without fictitious bidders

Efficiency Without fictitious bidders

Stability

Stability