Strongly Polynomial-Time Truthful Mechanisms in One Shot Paolo Penna 1, Guido Proietti 2, Peter...
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Transcript of Strongly Polynomial-Time Truthful Mechanisms in One Shot Paolo Penna 1, Guido Proietti 2, Peter...
Strongly Polynomial-Time Truthful Mechanisms in
One ShotPaolo Penna1, Guido Proietti2, Peter Widmayer3
1 Università di Salerno2 Università de l’Aquila3 ETH Zurich
Example: BGP Routing
An Autonomous System may report false link status to redirect traffic to another AS
AS1
AS2source destination
Link down
Networks, Protocols, Mechanisms
Network
Protocol “efficient communication”
Private Costs
Selfish agent
selected not selected
ti
cost ti cost 0
“bids” bi
Payments that incentivize agents to be truthfulTruthtelling is a dominant strategyReporting bi = ti maximizes the utility of agent i, always
utility = payment - cost
Networks, Protocols, Mechanisms
Network
Protocol “efficient communication”
Private Costs
Selfish agent
selected not selected
ti
cost ti cost 0
“bids” bi
“Efficient protocol” “Incentive compatible efficient protocol”
?“Efficient” Alg “Efficient” truthful mechanism (Alg,Pay)
? Algorithmic Mechanism Design [Nisan&Ronen’99]
< 3
Truthful Mechanisms
1 2
xShortest Path Tree
cheap expensivex
3
selected not selected
3 <
Depends on the “1” and “2”
agent bid
Monotone Algorithm
Truthful Mechanisms
Monotone algorithm Truthful mechanism
(Vickrey’61, Myerson’81)
Pay
0Tcheap expensivex
T
selected not selected
bid of agent ibids of other agents Alg
Truthful Mechanisms
Monotone algorithm Truthful mechanism
(Vickrey’61, Myerson’81)
Payments Thresholds
Truthful Mechanisms
(Alg,Pay)
Monotonicity Compute the payments
Techniques to solve both?
Efficient mechanism in one shot
Two algorithmic problemsnaive approach: weakly polynomial-time
Our (and Prior) Work
General technique for obtaining efficient mechanisms is one-shot
• write Alg as a “combination” of simpler algorithms
• compute the payments from the simpler algorithms
Monotonicity (easy to prove)Payment computations (efficient)
Prior work:• Monotone “combinations” [Mu’Alem&Nisan’02] • Fast payment computations for several “combinations” [Kao&Li&Wang’05]
compute payment no min-max problems
Limitations
Our (and Prior) Work
General technique for obtaining efficient mechanisms is one-shot
• write Alg as a “combination” of simpler algorithms
• compute the payments from the simpler algorithms
Mechanism for the Minimum Diameter Spanning Tree
max latency
Our (and Prior) Work
General technique for obtaining efficient mechanisms is one-shot
• write Alg as a “combination” of simpler algorithms
• compute the payments from the simpler algorithms
Mechanism for the Minimum Diameter Spanning Tree
• first mechanism, strongly polynomial-time (close to “best algorithm”)
MIN(A1,A2) : • run A1 and A2 independently;• choose the solution whose cost is smaller.
MIN(A1,A2):• compute X1 := A1(b) and X2:=A2(b)• if cost(X1,b) cost(X2,b) then return X1 else return X2
“Min combinations”
Objective function
MIN (A1,…,Ak) := MIN(A1,MIN(A2,…,Ak))
[Mu’Alem&Nisan’02]
Agents’ bids
Example: Minimum Radius Spanning Tree
Rooted tree minimizing the longest path to the root(locate a server and minimize the maximum latency)
1 1
x
x
T1
SPT2
T2
1 20
1
SPT1
MIN(SPT1,SPT2)
SPT2SPT1 SPT2
MRST
MIN(SPT2,SPT1)
Order matters!!
Different payments!Cost = height
Example: Minimum Radius Spanning Tree
Payment Computations (Idea)
x
A1
A2
MIN (A1,A2)MIN (A2,A1)
A1A2
The order
matters
T1R1L1TT
non-decreasing constant
Ah
Aj
Payment Computations (Idea)
x
The order
matters
RhLh
lowest
MIN(A1,…, Aj, …Ak)
Ah Ah
Threshold T = “leftmost ’’
General Technique
Alg is MIN-reducible in time
x
Ajnon-decreasing constant
Tj
Alg = MIN(A1,…,Ak)
Lj Rj
General Technique
Alg is MIN-reducible in time
Truthful (Alg,Pay) running in O(timeAlg + ) time
Main Application: MDST
MDST
Truthful mechnism for MDST running inO(n•(n,m)•timeMDST) time
Easy
Hard
Min Diam. Spanning Tree (Idea)
MDST = MIN(SPTe1,…,SPTem)
SPT
e1
Cost = height (max dist to every other node)
x
L R[Assin&Tamir’95]
Min Diam. Spanning Tree (Idea)
MDST = MIN(SPTe1,…,SPTem)
x
u
e1
SPT max x
u’ u’’ xL R
limit
= “at least limit”
LR
= “larger than limit”
Extensions• Technique: Every “Binary Game” (selected/not selected) • Minimum Radius Spanning Tree (better running time)• p-center (1-center MRST)
Open
• Approximation: NP-hard problems• MIN-reducible APX Alg?
• More general Agents (e.g. two edges per agent)