# Complexity Issues in Multiagent Resource Allocation Paul E

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Complexity Issues in Multiagent Resource Allocation Paul E. Dunne Dept. of Computer Science University of Liverpool United Kingdom 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 2

Modelling Resource Allocation A = {a 1 , … , an } – set of n agents. R = {r 1 , … , rm } – resource collection. U = {u 1 , … , un } – utility functions. Utility function – u – maps subsets of R to rational values. • An allocation is a partition of R into n sets - P = < P 1 ; … ; P n > • n, m denotes the set of allocations. • • 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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 ). 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 4

Assessing Allocations • Qualitative measures. Pareto Optimality Envy Freeness • Quantitative measures. Utilitarian Social Welfare Egalitarian Social Welfare 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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 ). 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 7

Quantitative Assessment • Utilitarian Social Welfare - u(P) = ui(Pi ) • Egalitarian Social Welfare - e(P) = min {ui(Pi ) } • One aim is to find allocations that maximise these. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 8

Complexity Considerations • Formulating decision problems. • Representing instances of such decision problems. • An important issue being how the collection {u 1 , … , un } is described. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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)? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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) 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 15

Complexity – Qualitative Case • ENVY-FREE is NP-complete with SLP and 2 agents. • PARETO OPTIMAL is co. NP-complete with 2 agents in both SLP and 2 -additive utility functions 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 16

Complexity – Quantitative Case • In 2 agent settings using SLP or 2 additive utility functions: WELFARE OPTIMISATION is NP-complete WELFARE IMPROVEMENT is NP-complete 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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>? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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” 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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) 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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? 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st 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. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 28

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