1 Foundations of Distributed Algorithmic Mechanism Design Joan Feigenbaum Yale University jf.
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1
Foundations of Distributed Algorithmic Mechanism Design
Joan FeigenbaumYale University
http://www.cs.yale.edu/~jf
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Two Views of Multi-agent Systems
CS ECON
Focus is on Computational & Communication Efficiency
Agents are Obedient, Faulty, or Byzantine
Focus is on Incentives
Agents are Strategic
3
Secure, Multiparty Function Evaluation
. . .
t 1
t 2
t 3t
n-1
t n
O = O (t 1, …, t
n)
• Each i learns O.• No i can learn anything about t
j
(except what he can infer from t i and O).
4
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 obedient agents learn O .
− No byzantine agent learns O or t
j for an obedient j.
5
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”
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Outline
• DAMD definitions and notation
• Short example: Multicast cost sharing
• General open questions
• Long example: Interdomain routing
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Definitions and Notation
t 1 t
n
Agent 1 Agent n
Mechanism
. . .
p 1 a
1 p na
n
O(Private) types: t
1, …, t n
Strategies: a 1, …, a
n
Payments: p i = p
i (a
1, …, a n)
Output: O = O (a 1, …, a
n)
Valuations: v i = v
i (t
i, O)
Utilities: u i = v
i + p
i
(Choose a i to maximize u
i.)
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“Strategyproof” Mechanism
For all i, t i, a
i, and a –i = (a
1, …, a i-1, a
i+i, …, a n)
v i
(t i, O (a
–i, t i)) + p
i (a
–i, t i)
v i
(t i, O (a
–i, a i)) + p
i (a
–i, a i)
• “Truthfulness”• “Dominant Strategy Solution Concept”
Nisan-Ronen ’99: Polynomial time O ( ) and p
i ( )
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Example: Task Allocation
Input: Tasks z1, …, zk
Agent i ’s type: t = (t1, …, t
k )
(tj is the minimum time in which i can complete z
j)
Feasible outputs: Z = Z 1 Z
2 … Z n
(Z i is the set of tasks assigned to agent i)
Valuations: v i
(t , Z) = − tj
Goal: Minimize max tj
i
i
i
i
i
i
zj Z
i
zj Z
iiZ
i
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Nisan-Ronen Min-Work Mechanism
O (a1, …, an ): Assign zj to agent with smallest aj
p i (a
1, …, a n ) = min a
j
[NR99]: Strategyproof, n-Approximation
Open Questions:
• Average case(s)?
• Distributed algorithm for Min-Work?
i
i’i i’zj Z
i
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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|| ).
“Good network complexity”
j=1
m
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Multicast Cost SharingMechanism-Design Problem
3 3
1 5 25
1,2 3,0
1,26,710
Users’ typesLink costs
Source
Which users receive the multicast?
Receiver Set
Cost Shares
How much does each receiver pay?
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Two Natural Mechanisms
Group-strategyproof
Budget-balanced
Minimum worst-case efficiency loss
Bad network complexity
Strategyproof
Efficient
Good network complexity
Shapley value
Marginal cost
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Marginal Cost
Receiver set: R* = arg max NW(R)R
NW(R) t i – C(T(R))
i R
Cost shares:
p i =
0 if i R*
t i – [NW( R*( t ) ) – NW( R*( t |i 0) )] o.w. {
Computable with two (short) messages per linkand two (simple) calculations per node. [FPS ’00]
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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 t i p
i , for all i R*
Any distributed algorithm that computes it mustsend (n) bits over (|T |) links in the worst case. [FKSS ’02]
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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.
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Open Question: More (Realistic)Distributed Algorithmic Mechanisms
Caching
P2P file sharing
Interdomain Routing
Distributed Task Allocation
Overlay Networks
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Open Question: Strategic Modeling
In each DAMD problem, which agents are
Obedient
Strategic
[ Byzantine ]
[ Faulty ] ?
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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]
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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
FOR i 1 to n
SIMULATE i to COMPUTE a i
O O (a 1, …, a
n); p p (a 1, …, a
n)
p 1 t
1 p nt
n
ONote: Loss of privacy Shift of computational load
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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?
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Open Question: Can IndirectMechanisms be More Efficient w.r.tComputation or Communication?
• Mechanism computation
• Agent computation
• Communication
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Interdomain-RoutingMechanism-design Problem
Inputs: Transit costs
Outputs: Routes, Payments
Qwest
Sprint
UUNET
WorldNet
Agents: Transit ASs
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Problem Statement
• Strategyproofness• “BGP-based” distributed algorithm
• Lowest-cost paths (LCPs)
Per-packet costs {ck } Agents’ types:
{route(i, j)}Outputs:
(Unknown) global parameter: Traffic matrix [Tij]
{pk}Payments:
Objectives:
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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
Hershberger-Suri, 2001
pl dG|cl= - dG|cl=0
• Compute m payments as quickly as 1
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Our Formulation vs. NR, HS
• Nodes, not links, are the strategic agents.
• All (source, destination) pairs
• Distributed “BGP-based” algorithm
More realistic model
Advantages:
Deployable via small changes to BGP
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Notation
• LCPs described by indicator function:
1 if k is on the LCP from i to j, when cost vector is c 0 otherwise
• c Ι (c1, c2, … ,, …, cn)
{Ik(c; i,j)
k
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A Unique VCG Mechanism
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
= ck Ik(c; i, j) + [ Ir(c Ι ; i, j ) cr - Ir(c; i, j ) cr ]
Theorem 1:
pijk
r r
i,j
pijk k
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Features of this Mechanism
• Payments have a very simple dependence on traffic Tij :
payment pk is the sum of per-packet prices .
• Price is 0 if k is not on LCP between i, j.
• Cost ck 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).
pijk
pijk
pijk
pijk
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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:
Entire paths are stored, not just next hop.
Dest. LCP LCP cost
AS3 AS5 3AS1AS1
AS7 AS2 2AS2
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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:
Receive routes from neighbors
Update routing table
Advertise modified routes
• Complexity measures: − Number of stages required for convergence − Total communication
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Towards Distributed Price Computation
= ck + Cost ( P-k(c; i,j) ) – Cost ( P(c; i,j) )
• LCPs to destination j form a tree
• Use data from i ’s neighbors a,b,d to compute
tree edge
non-tree edge
j
a
bdi
.
pijk
pijk
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Constructing k-avoiding Paths
Three possible cases for P-k(c; i, j):
j
a
bdi
k
i ’s neighbor on the path is (a) parent (b) child (d) unrelated
In each case, a relation to neighbor’s LCP or price, e.g., (b) = + cb + ci
is the minimum of these values.
pijk pbj
k
pijk
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A “BGP-based” Algorithm
AS3 AS5
c(i,1) AS1 c1
Dest. cost LCP and path prices LCP cost
AS1
• 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.pijk
pi1 3 pi1
5
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Performance of Algorithm
d’ = maxi,j,k || P-k ( c; i, j ) ||
d = maxi,j || P ( c; i, j ) ||
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).
Theorem 2:
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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?
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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 : 1.44Maximum node price: 990% of prices were 1 or 2
How do VCG prices interact with AS-graph formation?
Overcharging is not a major problem!
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