On Maximal Classes of Utility Functions for Efficient resource-at-a-time Negotiation

1. On Maximal Classes of Utility Functions for Efficient resource-at-a-time Negotiation Yann Chevaleyre, LAMSADE University of Paris 9 - Dauphine

2. MARA…the setting • Allocation of resources r1…rm among agents a1…an • Each agent’s preference is modeled with a utility function ui(R) • Social welfare of an allocation A is measured by sw(A) = i ui(R)

3. Problematic • If we restrict to one-resource-at-a-time deals (1-deals)What kind of utility function guarantees us to reach optimal social welfare ? • Definition: conditions on u1…un are said to permit 1-deal negotiation iff any sequence of IR 1-deal eventually results in an optimal allocation. Let’s define this

4. Outline • Modular Functions • def • properties • Negotiating with side-payments (w.s.p) • Sufficiency result • Sufficiency, necessity, maximality • Maximality result • Negotiating without side-payments (w/o.s.p) • Like-it-or-not functions • Sufficiency result + Maximality result

5. Modular utility functions • Intuition: linear utility function with possibly u()0 • Definition: • The class of modular function is noted M u’(R)

6. Useful later • Thus, u  M iff R,r1,r2 Properties of Modular Functions • if u()=0 then modular=linear • u is modular iff R1, R2 • Equivalently, u is modular iff R,r1,r2

7. Modularity permits 1-deal negotiation wsp • Lemma:A deal with side-payments is IR iff it increases social welfare • Theorem (follows AAMAS03): If u1…unM then 1-deal nego w.s.p is permitted. • The number of allocations is finite,we only need to show that if A is sub-optimal, then there always exists a IR deal (thus increasing sw by former lemma)

8. Modularity permits 1-deal negotiation wsp (proof) • Idea of the proof (example using 2 agents) • consider the following allocation Asub • consider the opt allocation Aopt r1 r2 r3 … rm agent 1 agent 2 sw(Asub) =u1()+u2()+ u1’(r1) + u2’(r2) + u2’(r3)+ … + u1’(rm) r1 r2 r3 … rm agent 1 agent 2 sw(Aopt) =u1()+u2()+ u2’(r1) + u1’(r2) + u2’(r3)+ … + u2’(rm)

9. Modularity permits 1-deal negotiation wsp (proof cont’d) • Because A is suboptimalsw(Asub) < sw(Aopt) • either u1’(r1)<u2’(r1)in which case moving r1 is IR • either u2’(r2)<u1’(r2) in which case moving r2 is IR • either … sw(Asub) =u1()+u2()+ u1’(r1) + u2’(r2) + u2’(r3)+ … + u1’(rm) sw(Aopt) =u1()+u2()+ u2’(r1) + u1’(r2) + u2’(r3)+ … + u2’(rm) sw(Asub) =u1()+u2()+ u1’(r1) + u2’(r2) + u2’(r3)+ … + u1’(rm) sw(Aopt) =u1()+u2()+ u2’(r1) + u1’(r2) + u2’(r3)+ … + u2’(rm)

10. Sufficiency, necessity • Sufficient condition (shown in the previous slides)if u1…unM, then 1-deal nego wsp is permitted • Problem : can we find a nessary+sufficient cond of the form « 1-deal nego wsp is permitted iff u1…unF. »? • Answer: NO !!!!Because we can find C1 and C2 such that: • if u1…unC1, then 1-deal nego wsp is permitted • if u1…unC2, then 1-deal nego wsp is permitted • 1-deal nego w.s.p. is not always permittedif u1…unC1 C2 • class F should include C1 andC2 and thus cannot permit!!!

11. Necessity, maximality • Problem : can we find a nessary+sufficient result of the form « 1-deal nego wsp is permitted iff u1…un verifies a given condition » ? • ANSWER: maybe, but the condition won’t be simple, and verifying may require more than poly timeConjecture : with most compact representations (k-additive, SLP), it is NP-hard to determine wether <u1…un> permits 1-deal nego wspArgument: it is NP-hard to determine if there is a 1-deal sequence from A1 to A2 (Dunne’s theorem using SLP utilities) • MaximalityThere is no class F  M, such thatif u1…unF, then 1-deal nego w.s-p is permitted

12. Maximality • Theo: There is no class F  M, such thatif u1…unF, then 1-deal nego wsp is permitted • Idea of the proof:Consider any utility function u1 MWe will show that M{u1} does not permit… • More precisely: • Given u1 M • Find u2  M , find allocation Asuch that • A is sub-optimal • There is no IR-deal getting out of A

13. Maximality proof (1/2) • Let u1  M. Then • Consider the allocations in which agent 1 owns R, and r1,r2 are shared among both. • We can build u2 Msuch that …

14. Maximality proof (2/2) • More precisely, u2 is made such that • for all resources r R, u2(r) << 0 • for all resources r  R  {r1,r2}, u2(r) >> 0

15. Modular Functions • def • properties • Negotiating with side-payments (wsp) • Sufficiency result • Maximality result • Negotiating without side-payments (w/o.sp) • Sufficiency result • Maximality result

16. Sufficiency (w/o.s.p) • Theo [AAMAS03]: if all utilities are 0-1 valued then 1-deals w/o.sp permits nego • Ex: • u1 = r1 + r4 + r5 • u2 = r1 + r2 + r3 + r6

17. Like-it-or-not functions • Let us associate to each ri two values • i (degree of satisfaction when holding the resource) • i (degree of unsatisfaction). • Each agent can either like a resource, dislike it, or be indifferent to it • Example: with 3 resources • u1 = r1 +5.r2 + 5.r3 • u2 = -3.r1 -3.r2 - 2.r3 • u3 = r1 - 2.r3 • These are like-it-or-not functions

18. Like-it-or-not functions (cont’d) • Notation: • given two vectors =(1… m),=(1… m) • the class M, denotes all like-it-or-not functions with parameters ,. • Note 1:M,  M • Note 2: 0-1 valued functions = M, with =(1,…1), and =(0,…0).

19. Sufficiency & Maximality • Theo: Given two vectors , • (sufficiency) if u1…unM, then 1-deal negotiation w/o.sp is permitted • Proof : same principle as for 0-1 valued functions • (maximality) There is no class F  M, , such that if u1…unF, then 1-deal nego w/o.sp is permitted • Proof : too long

20. Conclusion • Sufficiency result: • Slightly more general in the wsp case • Like-it-or-not : interesting new class for w/o.sp case • Future work: • Other classes also sufficient+maximal ? • Properties on the set of all sufficient+maximal classes ? • NP-completeness of verifying wether a utility profile permits 1-deal negotiation • Relaxation : notion of « quasi-permitness »