Transcript On Concise Encodings of Preferred Extensions

Complexity Issues in Multiagent
Resource Allocation
Paul E. Dunne
Dept. of Computer Science
University of Liverpool
United Kingdom
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Overview
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2.
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Modelling resource allocation.
Assessing allocations.
Complexity considerations
Computational complexity properties.
A Model for negotiating allocations
and its properties.
Open questions and conjectures
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Modelling Resource Allocation
A = {a1 , … , an } – set of n agents.
R = {r1 , … , rm } – resource collection.
U = {u1 , … , un } – utility functions.
Utility function – u – maps subsets of R
to rational values.
• An allocation is a partition of R into n
sets - P = <P1 ; … ; Pn > •
•
•
•
• n,m denotes the set of all allocations.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Assumptions
• Exactly one agent owns any resource,
i.e. R is non-shareable.
• Utility functions have no allocative
externality, i.e. for any P, Q  n,m with
Pi = Qi it holds that ui(Pi ) = ui(Qi ).
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Assessing Allocations
• Qualitative measures.
Pareto Optimality
Envy Freeness
• Quantitative measures.
Utilitarian Social Welfare
Egalitarian Social Welfare
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Qualitative Assessment I
• An allocation, P, is Pareto Optimal if for
every allocation, Q, that differs from it
should there be an agent for whom
ui(Qi ) > ui(Pi )
then there is another agent for whom
ui(Pi ) > ui(Qi ).
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Qualitative Assessment II
• An allocation, P, is Envy Free if no
agent assigns greater utility to the
resource set allocated to another agent
within P than it attaches to its own
allocation under P.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Quantitative Assessment
• Utilitarian Social Welfare - u(P)
u(P) =  ui(Pi )
• Egalitarian Social Welfare - e(P)
e(P) = min {ui(Pi ) }
• One aim is to find allocations that
maximise these.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Complexity Considerations
• Formulating decision problems.
• Representing instances of such
decision problems.
• An important issue being how the
collection {u1 , … , un } is described.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Some decision problems I
• ENVY-FREE
Instance: <A,R,U>
Question: Is there an envy-free allocation of R?
• PARETO OPTIMAL
Instance: <A,R,U> ; P  n,m
Question: Is P Pareto Optimal?
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28th February – 1st March 2005
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Some decision problems II
• WELFARE OPTIMISATION
Instance: <A,R,U>; K rational value.
Question: Is there an allocation with u(P)  K ?
• WELFARE IMPROVEMENT
Instance: <A,R,U>; P  n,m
Question: Is there Q  n,m with u(Q)> u(P)?
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Representing Utility Functions
• Possible options
Enumerate non-zero valued subsets of R
(‘bundle’ form)
Algorithm that computes u(S) given S
(‘program’ form)
Suitable algebraic formula, e.g.
u(S) = TR : |T|k (T)IS(T)
(‘k-additive’ form)
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Pros and Cons
• Bundle form – ‘easy’ to encode but
length of encoding could be exponential
in m.
• k-additive form – succinct for constant k
but not always possible.
• Program form – can be succinct; problem
Program run-time and termination
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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‘Suitable’ Program Form: SLP
• Straight-Line Programs –
m input bits encode subset S
t program lines – vr := vb  vd – b, d < r
• Can describe as m+t triples <r,b,d>.
• Poly-time computable u  poly. length SLP
• SLP for u can always be defined.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Complexity and Representation
• The form chosen to represent U has
little effect on the complexity of the
decision problems introduced earlier.
• Similarly, many results apply even when
only two agent settings are used.
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28th February – 1st March 2005
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Complexity – Qualitative Case
• ENVY-FREE is NP-complete with SLP
and 2 agents.
• PARETO OPTIMAL is coNP-complete with
2 agents in both SLP and 2-additive
utility functions
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Complexity – Quantitative Case
• In 2 agent settings using SLP or 2additive utility functions:
WELFARE OPTIMISATION is NP-complete
WELFARE IMPROVEMENT is NP-complete
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Negotiation Models
• With <A,R,U> there are |A||R| allocations.
• For P and Q distinct allocations, the deal
=<P,Q> replaces the allocation P with the
the allocation Q.
• It is not necessary for every agent to be given
a new allocation within a deal - A denotes the
set of agents whose allocation is changed by
implementing the deal.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Reducing the number of deals
• It is not feasible to review every deal.
• 2 methods to restrict the number of
deals in the search space:
Structural restrictions
Rationality restrictions
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Structural Restrictions
• Limit deals to those in which the number
of participating agents is bounded
and/or the number of resources
exchanged is bounded, e.g.
One resource-at-a-time (O-contract)
(at most) k-resources-at-at-time (C(k)-contract)
Exchange (or swap) contracts
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Rationality Restrictions
• Limit deals to those which “improve” an
agent’s view of its allocation, e.g.
Individual Rationality (IR) deals
<P,Q> is said to be IR if u(Q)> u(P)
• Thus, each agent places greater value on a
‘new’ allocation or (if it loses value) can be
‘compensated’ for its loss.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Problems with combined restrictions
• Assume <P,Q> is IR.
• <P,Q> is always realisable by a
sequence of O-contracts.
• <P,Q> is not always realisable by a
sequence of IR O-contracts.
• Similarly, replacing O-contracts by C(k)contract.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Associated decision problems
• IRO PATH
Instance: <A,R,U> ; IR deal <P,Q>
Question: Is there a sequence of IR O-contracts
implementing <P,Q>?
• IR(k) PATH
Instance: <A,R,U> ; IR deal <P,Q>
Question: Is there a sequence of IR C(k)contracts implementing <P,Q>?
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Complexity Properties
• In SLP model
IRO PATH is NP-hard
IR(k) PATH is NP-hard  k (constant)
IR(k) PATH is NP-hard for k=c.|R| with c0.5
• There are difficulties with establishing
membership in NP using the “obvious”
algorithm, i.e. “guess a path and check its
correctness”
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Length of IR O-contract paths
• Any deal <P,Q> can be implemented by
a sequence of at most |R| O-contracts.
• There are IR deals <P,Q> that can be
implemented by a sequence of IR Ocontracts but the shortest such
sequence has length
(2|R|) – (arbitrary U)
(2|R|/2) – (monotone U)
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
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Some Open Questions I
• Using 2-additive utility functions:
Complexity of ENVY-FREE?
Complexity of IRO PATH?
• Worst-case length of shortest IR O-contract
sequence for k-additive utility functions
• Upper bounds on complexity of IRO PATH,
noting that IRO PATHNP? is non-trivial.
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28th February – 1st March 2005
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Some Open Questions II
• Suppose the requirement for every deal to be
an IR O-contract is relaxed? e.g. by allowing
a “small” number of “irrational” deals and/or
deals which are not O-contracts.
Approximation algorithms
Do exponential length paths occur when t
irrational deals are allowed, with the same
deal having poly. length with t+1 irrational
deals?
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28th February – 1st March 2005
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Bibliography
• P.E. Dunne, M. Wooldridge & M. Laurence.
The Complexity of Contract Negotiation.
Artificial Intelligence, 2005 (in press)
• P.E. Dunne.
Extremal Behaviour in Multiagent Contract Negotiation.
Jnl. of Artificial Intelligence Res., 23, (2005), 41-78
Context dependence in mulitagent resource allocation.
• Y. Chevaleyre, U. Endriss, S. Estivie, & N. Maudet.
Multiagent resource allocation in k-additive domains:
preference representation and complexity.
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28th February – 1st March 2005
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