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Foundations of Distributed Algorithmic Mechanism Design

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- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

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( )

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 maxtj

i

i

i

i

i

i

zjZi

i

zjZi

Z

i

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

i

pi(a1, …, an) = min aj

i’

zjZi

i i’

[NR99]: Strategyproof, n-Approximation

- Open Questions:
- Average case(s)?
- Distributed algorithm for Min-Work?

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”

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

- Marginal cost

- Strategyproof
- Efficient
- Good network complexity

- Shapley value

- Group-strategyproof
- Budget-balanced
- Minimum worst-case efficiency loss
- Bad network complexity

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

R

NW(R) ti –C(T(R))

iR

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

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]

“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.

Caching

P2P file sharing

Interdomain Routing

Distributed Task Allocation

Overlay Networks

In each DAMD problem, which agents are

Obedient

Strategic

[ Byzantine ]

[ Faulty ] ?

- 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

Shift of computational load

- 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?

- Mechanism computation
- Agent computation
- Communication

WorldNet

Qwest

UUNET

Sprint

Agents: Transit ASs

Inputs:Transit costs

Outputs: Routes, Payments

Agents’ types:

Per-packet costs {ck}

Outputs:

{route(i, j)}

{pk}

Payments:

(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

Advantages:

- 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

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

k

pij

pij

pij

pij

- 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:

Advertise modified routes

Receive routes from neighbors

Update routing table

- Complexity measures:
− Number of stages required for convergence

- − Total communication

j

a

i

d

b

k

k

pij

pij

=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

k

pij

pij

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.

k

pij

3

pi1

5

pi1

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?