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Algorithmic Mechanism Design
Paolo Penna
Game Theory
Game Theory
Ideally
Game Theory
In practice…
Ideally
The Internet
No central authority, open, self organized, anarchic
The Internet
No central authority, open, self organized, anarchic
Different “components” which• have their own goal
• may not follow the “protocol”
The Internet
No central authority, open, self organized, anarchic
Different “components” which• have their own goal
• may not follow the “protocol”
Selfish agents
The Internet
No central authority, open, self organized, anarchic
AS1
AS2source
destination
The Internet
No central authority, open, self organized, anarchic
AS1
AS2source
destination
Congestion
The Internet
No central authority, open, self organized, anarchic
AS1
AS2source
destination
Congestion
The Internet
No central authority, open, self organized, anarchic
An Autonomous System may report false link status
to redirect traffic to another AS
AS1
AS2source
destination
Congestion
A Toy Problem
s d
2
10
A Toy Problem
s d
2
10
private costs
A Toy Problem
s d
2
10
A Toy Problem
s d
2
10
100
A Toy Problem
s d
2
10
Payment= reported cost
A Toy Problem
s d
2
10
Payment= reported cost
9.999
A Toy Problem
s d
2
10
Payment= 2nd best cost
9.999
A Toy Problem
s d
2
10
Payment= 2nd best cost
A Toy Problem
s d
2
10
Payment= 2nd best cost
utility(“truth”) = 10 – 2 ≥ utility(“false”)
Truthful Mechanism
A Toy Problem
s d
2
10
Truthful Mechanism(Vickerey'61 – Clarke'71 – Groves'73)
…minimize the sum of all costs
Shortest Path, Min Spanning Tree, Steiner Tree,...
“Good” Protocol
1. Optimal Solution...
...if agents do not cheat
2. Truthful (no reason to cheat)
“Good” Protocol
1. Optimal Solution...
...if agents do not cheat
2. Truthful (no reason to cheat)
Algorithms
“Good” Protocol
1. Optimal Solution...
...if agents do not cheat
2. Truthful (no reason to cheat)
Algorithms
Game Theory
“Good” Protocol
1. Optimal Solution...
...if agents do not cheat
2. Truthful (no reason to cheat)
Algorithms
Game Theory
Not always possible
Makespan in selfish unrelated machines (Nisan&Ronen'99)
Truthful MechanismsTruthful
mechanism(Alg, Pay)
Monotonealgorithm
Alg
(Myerson'81, Rochet'87,...,Bikhchandani et al'06, Saks&Yu'05,...)
Truthful Mechanisms
s d
t1
t2
selectedi
ti
Truthfulmechanism(Alg, Pay)
Monotonealgorithm
Alg
(Myerson'81, Rochet'87,...,Bikhchandani et al'06, Saks&Yu'05,...)
Truthful Mechanisms
s d
t1
t2
selectedi
ti
worki
s d
t1
t2
ti
Truthfulmechanism(Alg, Pay)
Monotonealgorithm
Alg
(Myerson'81, Rochet'87,...,Bikhchandani et al'06, Saks&Yu'05,...)
Truthful Mechanisms
s d
t1
t2
selectedi
ti
worki
s d
t1
t2
ti
...
generalcosts
suitableinequality
Truthfulmechanism(Alg, Pay)
Monotonealgorithm
Alg
(Myerson'81, Rochet'87,...,Bikhchandani et al'06, Saks&Yu'05,...)
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
Greedy:
1+ℇ 1
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
3
Greedy:
1+ℇ 1
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
32
Greedy:
1+ℇ 1
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
32
2
Greedy:
1+ℇ 1
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
32
2
Greedy:
1+ℇ 1 (1+ℇ)2
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
32
2
1+ℇ1
Greedy:
1+ℇ 1 (1+ℇ)2
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
32
2
2
2
3
1+ℇ1
Greedy:
1+ℇ 1 (1+ℇ)2
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
32
2
2
2
3
1+ℇ1
Greedy:
1+ℇ 1 (1+ℇ)2
not monotone
Monotone Algorithms?
worki
s d3
2
2
t1
ti
t2
Redesign approximation algorithms:
PTAS or FPTAS for related machines (...,Andelman et al'07, Christodoulou&Kovacs'10)
Private cost + private capacity
Simple characterization of truthfulness(Auletta&Penna&Persiano'09)
2
2
speed, capacity
Private cost + private capacity
Simple characterization of truthfulness(Auletta&Penna&Persiano'09)
2
2
speed, capacity
Private cost + private capacity
Simple characterization of truthfulness(Auletta&Penna&Persiano'09)
2
2
cost, capacity
speed, capacity
Private cost + private capacity
Simple characterization of truthfulness(Auletta&Penna&Persiano'09)
2
2
cost, capacity
speed, capacity
or
Multi Parameter Problems
Truthfulness with standard techniques: Exact exp time or Polytime unbounded APX
(Ambuehl&Clementi&Penna&Rossi&Silvestri'03)
Multi Parameter Problems
Truthfulness with standard techniques: Exact exp time or Polytime unbounded APX
(Ambuehl&Clementi&Penna&Rossi&Silvestri'03)
Multi Parameter Problems
Connectivity +
Min Total Energy
Save Energy
Truthfulness with standard techniques: Exact exp time or Polytime unbounded APX
(Ambuehl&Clementi&Penna&Rossi&Silvestri'03)
Multi Parameter Problems
Connectivity +
Min Total Energy
Save Energy
Truthfulness with standard techniques: Exact exp time or Polytime unbounded APX
(Ambuehl&Clementi&Penna&Rossi&Silvestri'03)
s d
2
10
Payments: 2nd best
s d
2
10
Payments: 2nd best
s d
2
101000
Payments: 2nd best
Collusion
s d
2
101000
No “useful” mechanism is collusion-resistant(Schummer'00)
Payments: 2nd best
Collusion
s d
2
10
More optimization problems + collusion resistant(Penna&Ventre'08 '09)
Partial Verification:
Possible Approaches
Collusion
s d
2
10
More optimization problems + collusion resistant(Penna&Ventre'08 '09)
Partial Verification:
Possible Approaches
1
Collusion
s d
2
10
More optimization problems + collusion resistant(Penna&Ventre'08 '09)
Partial Verification:
Possible Approaches
1
Collusion
s d
2
10
More optimization problems + collusion resistant(Penna&Ventre'08 '09)
Partial Verification:
Possible Approaches
1
Delay > 1
Collusion
s d
t1
t2
Two-values: optimization is possible(Mihalak,Penna,Widmayer,14+)
Restricted Types: ti ∈ {Low, High}
Possible Approaches
Remarks
Optimal Solutions
Incentive CompatibleAlgorithmic Properties
Non trivial to combine (compromise)● how much do we loose?● “altruism” is valuable resource
Thank You
