Communication NetworksA Second Course

Jean WalrandDepartment of EECSUniversity of California at Berkeley

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. 40th Annual Conference

on Information Sciences and Systems (CISS06), Princeton, NJ, Mar 22-24, 2006. .

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 one-

dimensional bids for strategic users

Problem (continued)

Necessary and sufficient conditions

Mechanism

“bids”

Kelly Mechanism

Network problem

Solution:

User problem

Kelly Mechanism (continued)

fi(x) = log(xi) surrogate valuation: Wi(x) = wilog(xi)

one-dimensional signals: wi instead of Ui(x)

Problems with strategic playersUser 0

User 1 User 2

5 8U0(x) = x

U1(x) = x U2(x) = x/2

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 players2) NE exists, it may be very inefficient:

User 0

User 1 User 2 User 3 User L

1 1 1 1 1

Ui(x) = x, i = 1, …, L

U0(x) = x, 0 < < L

Social: x0 = 0, xi = 1, I = 1, …, LNE: x0 = ( + L)-1, xi = ( + 1)-1

Loss: (2 + L)[L( + 1)]-1

Let 2 = L and let L

VCG Mechanism

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

VCG Mechanism (continued)

The bids are functions difficult to implement!

VCG-Kelly Mechanism

VCG-Kelly Mechanism (continued)

Efficiency

Without fictitious bidders

Stability