A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Clemens...
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A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Clemens Thielen Stephan Westphal 3rd International Workshop on Computational Social Choice 15 September 2010 [email protected]
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Transcript of A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Clemens...
A Combinatorial Algorithm forStrong Implementation of Social Choice Functions
Clemens Thielen Stephan Westphal
3rd International Workshop on Computational Social Choice
15 September 2010
Problem DefinitionSocial choice setting with private information:
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n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Problem Definition (2)We saw (previous talk):
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n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Our Results•
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n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
System of InequalitiesImplementation of a social choice function :
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Weak Implementation
Strong Implementation
Node Potential Interpretation
System can be interpreted as finding a node potential:
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
constant
constant
Characterization (Weak Implementation)
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n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
Theorem:[Gui et al. 2005]
Characterization (Strong Implementation)
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
Theorem:[this paper]
The Algorithm
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
For weak implementation:
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For strong implementation:
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Strict inequalitiesin the system
must be fullfilled.
Perturbation of Node Potentials
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
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Perturbation in Graph Gi
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
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Slack of arc (x,x
´)
Slack of incoming
arcs becomes positive
Finding the Node x
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
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You shrink it!
Summary of Results
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
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Thank you!
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
