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

2. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 2 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

3. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 3 Modelling Resource Allocation A = {a 1, …, a n } – set of n agents. R = {r 1, …, r m } – resource collection. U = {u 1, …, u n } – utility functions. Utility function – u – maps subsets of R to rational values. An allocation is a partition of R into n sets - P = -  n,m denotes the set of all allocations.

4. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 4 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 P i = Q i it holds that u i (P i ) = u i (Q i ).

5. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 5 Assessing Allocations Qualitative measures. Pareto Optimality Envy Freeness Quantitative measures. Utilitarian Social Welfare Egalitarian Social Welfare

6. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 6 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 u i (Q i ) > u i (P i ) then there is another agent for whom u i (P i ) > u i (Q i ).

7. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 7 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.

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

9. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 9 Complexity Considerations Formulating decision problems. Representing instances of such decision problems. An important issue being how the collection {u 1, …, u n } is described.

10. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 10 Some decision problems I ENVY-FREE Instance: Question: Is there an envy-free allocation of R? PARETO OPTIMAL Instance: ; P   n,m Question: Is P Pareto Optimal?

11. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 11 Some decision problems II WELFARE OPTIMISATION Instance: ; K rational value. Question: Is there an allocation with  u (P)  K ? WELFARE IMPROVEMENT Instance: ; P   n,m Question: Is there Q   n,m with  u (Q)>  u (P)?

12. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 12 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)I S (T) (‘k-additive’ form)

13. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 13 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

14. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 14 ‘Suitable’ Program Form: SLP Straight-Line Programs – m input bits encode subset S t program lines – v r := v b  v d – b, d < r Can describe as m+t triples. Poly-time computable u  poly. length SLP SLP for u can always be defined.

15. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 15 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.

16. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 16 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

17. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 17 Complexity – Quantitative Case In 2 agent settings using SLP or 2- additive utility functions: WELFARE OPTIMISATION is NP-complete WELFARE IMPROVEMENT is NP-complete

18. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 18 Negotiation Models With there are |A| |R| allocations. For P and Q distinct allocations, the deal  = 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.

19. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 19 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

20. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 20 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

21. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 21 Rationality Restrictions Limit deals to those which “improve” an agent’s view of its allocation, e.g. Individual Rationality (IR) deals 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.

22. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 22 Problems with combined restrictions Assume is IR. is always realisable by a sequence of O-contracts. is not always realisable by a sequence of IR O-contracts. Similarly, replacing O-contracts by C(k)- contract.

23. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 23 Associated decision problems IRO PATH Instance: ; IR deal Question: Is there a sequence of IR O-contracts implementing ? IR(k) PATH Instance: ; IR deal Question: Is there a sequence of IR C(k)- contracts implementing ?

24. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 24 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”

25. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 25 Length of IR O-contract paths Any deal can be implemented by a sequence of at most |R| O-contracts. There are IR deals 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)

26. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 26 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.

27. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 27 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?

28. 2 nd Agentlink III TFG-MARA, Ljubljana, Slovenia 28 th February – 1 st March 2005 28 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.