Foundations of distributed algorithmic mechanism design
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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

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Foundations of distributed algorithmic mechanism design

Foundations of Distributed Algorithmic Mechanism Design

Joan Feigenbaum

Yale University

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


Two views of multi agent systems

Two Views of Multi-agent Systems

CS

ECON

Focus is on

Incentives

Focus is on

Computational &

Communication

Efficiency

Agents are

Strategic

Agents are

Obedient,

Faulty, or

Byzantine


Secure multiparty function evaluation

. . .

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).


Smfe literature

SMFE Literature

  • 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.


Internet computation

Internet Computation

  • 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”


Outline

Outline

  • DAMD definitions and notation

  • Short example: Multicast cost sharing

  • General open questions

  • Long example: Interdomain routing


Definitions and notation

. . .

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.)


Strategyproof mechanism

“Strategyproof” Mechanism

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( )


Example task allocation

Example: Task Allocation

Input: Tasks z1, …, zk

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


Nisan ronen min work mechanism

Nisan-Ronen Min-Work Mechanism

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

Distributed AMD [FPS ’00]

Agents 1, …, n

Interconnection network T

Numerical input {c1, … cm}

  • O(|T|) messages total

  • O(1) messages per link

  • Polynomial time local computation

  • Maximum message size is

    polylog(n, |T |) and poly( ||cj|| ).

m

j=1

“Good network complexity”


Multicast cost sharing mechanism design problem

Multicast Cost SharingMechanism-Design Problem

Receiver Set

Source

Which users receive the multicast?

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

Link costs


Two natural mechanisms

Two Natural Mechanisms

  • Marginal cost

  • Strategyproof

  • Efficient

  • Good network complexity

  • Shapley value

  • Group-strategyproof

  • Budget-balanced

  • Minimum worst-case efficiency loss

  • Bad network complexity


Marginal cost

Marginal Cost

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]


Shapley value

Shapley Value

Cost shares: c(l) is shared equally by

all receivers downstream of l.

(Non-receivers pay 0.)

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]


Group strategyproof bb cost sharing

Group-Strategyproof, BB Cost Sharing

“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

Open Question: More (Realistic)Distributed Algorithmic Mechanisms

 Caching

 P2P file sharing

 Interdomain Routing

 Distributed Task Allocation

 Overlay Networks


Open question strategic modeling

Open Question: Strategic Modeling

In each DAMD problem, which agents are

 Obedient

 Strategic

[  Byzantine ]

[  Faulty ] ?


Open question what about provably hard damd problems

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]


Revelation principle

Revelation Principle

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

Shift of computational load


Open question design privacy preserving dams

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 indirect mechanisms be more efficient w r t computation or communication

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

  • Mechanism computation

  • Agent computation

  • Communication


Interdomain routing mechanism design problem

Interdomain-RoutingMechanism-design Problem

WorldNet

Qwest

UUNET

Sprint

Agents: Transit ASs

Inputs:Transit costs

Outputs: Routes, Payments


Problem statement

Agents’ types:

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


Previous work

Previous Work

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


Our formulation vs nr hs

Our Formulation vs. NR, HS

  • Nodes, not links, are the strategic agents.

  • All (source, destination) pairs

  • Distributed “BGP-based” algorithm

Advantages:

  • More realistic model

  • Deployable via small changes to BGP


Notation

Notation

  • 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


A unique vcg mechanism

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


Features of this mechanism

k

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).


Bgp based computational model 1

BGP-based Computational Model (1)

  • 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.


Bgp based computational model 2

BGP-based Computational Model (2)

  • 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:

Advertise modified routes

Receive routes from neighbors

Update routing table

  • Complexity measures:

    − Number of stages required for convergence

  • − Total communication


Towards distributed price computation

j

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


Constructing k avoiding paths

k

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.


A bgp based algorithm

k

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.


Performance of algorithm

Performance of Algorithm

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).


Open question strategy in computation

Open Question: Strategy in Computation

  • 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?


Open question overcharging

Open Question: Overcharging

  • 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?


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