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# Foundations of Distributed Algorithmic Mechanism Design - PowerPoint PPT Presentation

Foundations of Distributed Algorithmic Mechanism Design. Joan Feigenbaum Yale University. http://www.cs.yale.edu/~jf. Two Views of Multi-agent Systems. CS. ECON. Focus is on Incentives. Focus is on Computational & Communication Efficiency. Agents are Strategic. Agents are

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Foundations of Distributed Algorithmic Mechanism Design

Joan Feigenbaum

Yale University

http://www.cs.yale.edu/~jf

CS

ECON

Focus is on

Incentives

Focus is on

Computational &

Communication

Efficiency

Agents are

Strategic

Agents are

Obedient,

Faulty, or

Byzantine

tn-1

t3

tn

t2

t1

Secure, Multiparty Function Evaluation

O= O(t1, …, tn)

• Each i learns O.

• No i can learn anything about tj

• (except what he can infer from tiand O).

• Agents 1, … , n are obedient or byzantine.

• Obedient agents limited to probabilistic polynomial

• time. (Sometimes, byzantine agents too.)

• Typical assumption:

• at most n/3 byzantine agents

• Typical successful protocol:

− All obedientagents learn O.

• − No byzantineagent learns Oor

• tj for an obedientj.

• Both incentives and computational and

• communication efficiency matter.

• “Ownership, operation, and use by numerous

• independent self-interested parties give the

• Internet the characteristics of an economy as

• well as those of a computer.”

 DAMD: “Distributed Algorithmic Mechanism Design”

• DAMD definitions and notation

• Short example: Multicast cost sharing

• General open questions

• Long example: Interdomain routing

Agent 1

Agent n

p1

a1

an

pn

Mechanism

Definitions and Notation

t1

tn

O

Output: O = O(a1,…, an)

Valuations: vi = vi(ti, O)

Utilities: ui =vi +pi

(Private) types: t1, …, tn

Strategies: a1, …, an

Payments: pi = pi(a1, …, an)

(Choose ai to maximize ui.)

For all i, ti, ai, and a–i = (a1, …, ai-1, ai+i, …, an)

vi(ti,O(a–i,ti))+pi(a–i,ti)

vi(ti,O(a–i,ai))+pi(a–i,ai)

• “Truthfulness”

• “Dominant Strategy Solution Concept”

Nisan-Ronen ’99:

Polynomial time O( ) and pi( )

Agent i’s type: t = (t1, …, tk)

(tj is the minimum time in which i can complete zj)

Feasible outputs: Z = Z1 Z2 … Zn

(Zi is the set of tasks assigned to agent i)

Valuations: vi(t,Z) = − tj

Goal: Minimize maxtj

i

i

i

i

i

i

zjZi

i

zjZi

Z

i

O(a1, …, an): Assignzj to agent with smallest aj

i

pi(a1, …, an) =  min aj

i’

zjZi

i  i’

[NR99]: Strategyproof, n-Approximation

• Open Questions:

• Average case(s)?

• Distributed algorithm for Min-Work?

Distributed AMD [FPS ’00]

Agents 1, …, n

Interconnection network T

Numerical input {c1, … cm}

• O(|T|) messages total

• Polynomial time local computation

• Maximum message size is

polylog(n, |T |) and poly( ||cj|| ).

m

j=1

“Good network complexity”

Multicast Cost SharingMechanism-Design Problem

Source

3

3

1,2

3,0

1

5

5

2

Cost Shares

How much does each receiver pay?

10

6,7

1,2

Users’ types

• Marginal cost

• Strategyproof

• Efficient

• Good network complexity

• Shapley value

• Group-strategyproof

• Budget-balanced

• Minimum worst-case efficiency loss

Receiver set: R* = arg max NW(R)

R

NW(R) ti –C(T(R))

iR

Cost shares:

{

0 if i  R*

ti – [NW(R*( t)) – NW(R*( t |i0))] o.w.

pi =

Computable with two (short) messages per link

and two (simple) calculations per node. [FPS ’00]

Cost shares: c(l) is shared equally by

Receiver set: Biggest R* such that ti pi,

for all i  R*

Any distributed algorithm that computes it must

send (n) bits over (|T|) links in the worst case.

[FKSS ’02]

“Representative Hard Problem” to solve on the Internet

Hard if it must be solved:

• By a distributed algorithm

• Computationally efficiently

• Incentive compatibly

Becomes easy if one requirement is dropped.

Open Question: Find other representative hard problems.

Open Question: More (Realistic)Distributed Algorithmic Mechanisms

 Caching

 P2P file sharing

 Interdomain Routing

 Overlay Networks

In each DAMD problem, which agents are

 Obedient

 Strategic

[  Byzantine ]

[  Faulty ] ?

Open Question: What about “provably hard” DAMD problems?

• AMD approximation is subtle; can destroy

• strategyproofness

• “Feasibly dominant strategies” [NR ’01]

• “Strategically faithful” approximation [FKSS ’01]

• “Tolerable manipulability” [FKSS ’01]

If there is a mechanism (O, p) that implements a

design goal, then there is one that does so truthfully.

Agent 1

Agent n

. . .

p1

t1

tn

pn

FOR i 1 to n

SIMULATE i to COMPUTEai

O O(a1, …, an);p  p(a1, …, an)

O

Note: Loss of privacy

Open Question: Design Privacy-Preserving DAMs

• Cannot simply “compose” a DAM with a

• standard SMFE protocol.

−  (n) obedient agents

−Unacceptable network complexity

− Agents don’t “know” each other.

• New SMFE techniques?

Open Question: Can IndirectMechanisms be More Efficient w.r.tComputation or Communication?

• Mechanism computation

• Agent computation

• Communication

Interdomain-RoutingMechanism-design Problem

WorldNet

Qwest

UUNET

Sprint

Agents: Transit ASs

Inputs:Transit costs

Outputs: Routes, Payments

Per-packet costs {ck}

Outputs:

{route(i, j)}

{pk}

Payments:

Problem Statement

(Unknown) global parameter: Traffic matrix [Tij]

Objectives:

• Lowest-cost paths (LCPs)

• Strategyproofness

• “BGP-based” distributed algorithm

Nisan-Ronen, 1999

• Single (source, destination) pair

• Links are the strategic agents

• “Private type” of l is cl

• (Centralized) strategyproof, polynomial-time mechanism

pl dG|cl= - dG|cl=0

Hershberger-Suri, 2001

• Compute m payments as quickly as 1

• Nodes, not links, are the strategic agents.

• All (source, destination) pairs

• Distributed “BGP-based” algorithm

• More realistic model

• Deployable via small changes to BGP

• LCPs described by indicator function:

{

1 if k is on the LCP from i to j,

when cost vector is c

0 otherwise

Ik(c; i,j)

• cΙ  (c1, c2, … ,, …, cn)

k

pij

pij

k

k

A Unique VCG Mechanism

Theorem 1:

For a biconnected network, if LCP routes are always chosen, there is a unique strategyproof mechanism that gives no payment to nodes that carry no transit traffic. The payments are of the form

pk = Tij , where

i,j

= ckIk(c; i, j)+ [Ir(cΙ; i, j)cr -  Ir(c; i, j)cr]

k

r

r

k

k

k

pij

pij

pij

pij

Features of this Mechanism

• Payments have a very simple dependence on traffic Tij:

• payment pk is the sum of per-packet prices .

• Price is 0ifk is not on LCP between i, j.

• Costck is independent of i and j, but price

• depends on i and j.

• Price is determined by cost of min-cost path from

• i to j not passing through k (min-cost “k-avoiding” path).

• Follow abstract BGP model of Griffin and Wilfong:

Network is a graph with nodes corresponding to ASes and bidirectional links; intradomain-routing issues are ignored.

• Each AS has a routing table with LCPs to all other nodes:

LCP cost

LCP

Dest.

AS1

AS3

AS5

AS1

3

AS2

AS7

AS2

2

Entire paths are stored, not just next hop.

• An AS “advertises” its routes to its neighbors in

• the AS graph, whenever its routing table changes.

• The computation of a single node is an infinite

• sequence of stages:

Update routing table

• Complexity measures:

− Number of stages required for convergence

• − Total communication

a

i

d

b

k

k

pij

pij

Towards Distributed Price Computation

=ck+Cost(P-k(c; i,j))–Cost(P(c; i,j))

• LCPs to destination j form a tree

tree edge

non-tree edge

.

• Use data from i’s neighbors a,b,d to compute

k

pij

pij

Constructing k-avoiding Paths

Three possible cases for P-k(c; i, j):

j

i’s neighbor on the path is

(a) parent

(b) child

(d) unrelated

k

a

i

b

d

In each case, a relation to neighbor’s LCP or price, e.g.,

(b) =+cb+ci

k

pbj

• is the minimum of these values.

pij

3

pi1

5

pi1

A “BGP-based” Algorithm

LCP cost

LCP and path prices

Dest.

cost

AS3

AS5

AS1

c(i,1)

AS1

c1

• LCPs are computed and advertised to neighbors.

• Initially, all prices are set to .

• Each node repeats:

• − Receive LCP costs and path prices from neighbors.

• −Recompute path prices.

• − Advertise changed prices to neighbors.

Final state: Node i has accurate values.

d = maxi,j||P(c; i, j)||

d’ = maxi,j,k||P-k(c; i, j)||

Theorem 2:

Our algorithm computes the VCG prices correctly, uses routing tables of size O(nd) (a constant factor increase over BGP), and converges in at most (d+d’) stages (worst-case additive penalty of d’ stages over the BGP convergence time).

• Mechanism is strategyproof : ASes have no

• incentive to lie about ck’s.

• However, payments are computed by the strategic

• agents themselves.

• How do we reconcile the strategic model with the

• computational model?

• “Quick fix” : Digital Signatures

• [Mitchell, Sami, Talwar, Teague]

• Is there a way to do this without a PKI?

• In the worst case, path price can be arbitrarily

• higher than path cost [Archer&Tardos, 2002].

• This is a general problem wiith VCG mechanisms.

• Statistics from a real AS graph, with unit costs:

Mean node price : 1.44

Maximum node price: 9

90% of prices were 1 or 2

Overcharging is not a major problem!

How do VCG prices interact with AS-graph formation?