Complexity Issues in Multiagent Resource Allocation

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Complexity Issues in Multiagent Resource Allocation

Paul E. DunneDept. of Computer Science

University of LiverpoolUnited Kingdom

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Overview1. Modelling resource allocation.2. Assessing allocations.3. Complexity considerations4. Computational complexity

properties.5. A Model for negotiating allocations6. and its properties.7. Open questions and conjectures

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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.

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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 ).

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Assessing Allocations

• Qualitative measures.Pareto Optimality

Envy Freeness• Quantitative measures.

Utilitarian Social WelfareEgalitarian Social Welfare

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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 whomui(Pi ) > ui(Qi ).

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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.

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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.

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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.

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Some decision problems I

• ENVY-FREEInstance: <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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Some decision problems II

• WELFARE OPTIMISATIONInstance: <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)?

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Representing Utility Functions• Possible optionsEnumerate non-zero valued subsets of R

(‘bundle’ form)

Algorithm that computes u(S) given S (‘program’ form)

Suitable algebraic formula, e.g.

u(S) = TR : |T|k

(T)IS(T)

(‘k-additive’ form)

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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

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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.

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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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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

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Complexity – Quantitative Case• In 2 agent settings using SLP or 2-

additive utility functions:WELFARE OPTIMISATION is NP-complete

WELFARE IMPROVEMENT is NP-complete

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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.

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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 restrictionsRationality restrictions

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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

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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.

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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.

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Associated decision problems

• IRO PATHInstance: <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>?

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Complexity Properties

• In SLP modelIRO PATH is NP-hard

IR(k) PATH is NP-hard k (constant)IR(k) PATH is NP-hard for k=c.|R| with

c0.5• There are difficulties with establishing

membership in NP using the “obvious” algorithm, i.e. “guess a path and check its correctness”

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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 O-contracts but the shortest such sequence has length

(2|R|) – (arbitrary U) (2|R|/2) – (monotone U)

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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 PATHNP? is non-trivial.

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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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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), 41-78Context dependence in mulitagent resource allocation.• Y. Chevaleyre, U. Endriss, S. Estivie, & N. Maudet.Multiagent resource allocation in k-additive domains:

preference representation and complexity.