2nd Agentlink III TFG-MARA, Ljubljana, Slovenia

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Complexity Issues in Multiagent Resource
Allocation

• Paul E. Dunne
• Dept. of Computer Science
• University of Liverpool
• United Kingdom

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Overview

1. Modelling resource allocation.
2. Assessing allocations.
3. Complexity considerations
4. Computational complexity properties.
5. A Model for negotiating allocations
6. 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
  ltP1 Pn gt -
• ?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 Welfare
• Egalitarian 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 ) gt ui(Pi )
• then there is another agent for whom
• ui(Pi ) gt 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-FREE
• Instance ltA,R,Ugt
• Question Is there an envy-free allocation of R?
• PARETO OPTIMAL
• Instance ltA,R,Ugt P ? ?n,m
• Question Is P Pareto Optimal?

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Some decision problems II

• WELFARE OPTIMISATION
• Instance ltA,R,Ugt K rational value.
• Question Is there an allocation with ?u(P) ? K ?
• WELFARE IMPROVEMENT
• Instance ltA,R,Ugt P ? ?n,m
• Question Is there Q ? ?n,m with ?u(Q)gt ?u(P)?

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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)

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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 lt r
• Can describe as mt triples lt?r,b,dgt.
• 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 ltA,R,Ugt there are AR allocations.
• For P and Q distinct allocations, the deal
  ?ltP,Qgt 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 restrictions
• Rationality 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 agents
  view of its allocation, e.g.
• Individual Rationality (IR) deals
• ltP,Qgt is said to be IR if ?u(Q)gt ?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 ltP,Qgt is IR.
• ltP,Qgt is always realisable by a sequence of
  O-contracts.
• ltP,Qgt 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 PATH
• Instance ltA,R,Ugt IR deal ltP,Qgt
• Question Is there a sequence of IR O-contracts
  implementing ltP,Qgt?
• IR(k) PATH
• Instance ltA,R,Ugt IR deal ltP,Qgt
• Question Is there a sequence of IR
  C(k)-contracts implementing ltP,Qgt?

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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 kc.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

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Length of IR O-contract paths

• Any deal ltP,Qgt can be implemented by a sequence
  of at most R O-contracts.
• There are IR deals ltP,Qgt that can be implemented
  by a sequence of IR O-contracts but the shortest
  such sequence has length ?(2R)
  (arbitrary U) ?(2R/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 PATH?NP? 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 t1 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),
  41-78
• 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.