Download Complexity Issues in Multiagent Resource Allocation

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
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
1
Overview
1.
2.
3.
4.
5.
6.
7.
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
2
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
3
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
4
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
5
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
6
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
7
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
8
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
9
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?
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
10
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
11
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
12
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
13
‘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
14
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.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
15
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
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
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.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
26
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?
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
27
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.
2nd Agentlink III TFG-MARA, Ljubljana, Slovenia
28th February – 1st March 2005
28