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 r 1 …r m among agents a 1 …a n Each agent’s preference is modeled with a utility function u i (R) Social welfare of an allocation A is measured by sw(A) =  i u i (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 u 1 …u n 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 1. Modular Functions 1. def 2. properties 2. Negotiating with side-payments (w.s.p) 1. Sufficiency result 2. Sufficiency, necessity, maximality 3. Maximality result 3. Negotiating without side-payments (w/o.s.p) 1. Like-it-or-not functions 2. 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. Properties of Modular Functions if u(  )=0 then modular=linear u is modular iff  R 1, R 2 Equivalently, u is modular iff  R,r 1,r 2 Thus, u  M iff  R,r 1,r 2 Useful later

7. Modularity permits 1-deal negotiation wsp Lemma: A deal with side-payments is IR iff it increases social welfare Theorem (follows AAMAS03) : If u 1 …u n  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 A sub  consider the opt allocation A opt X…X XX… r 1 r 2 r 3 … r m agent 1 agent 2 sw(A sub ) =u 1 (  )+u 2 (  )+ u 1 ’(r 1 ) + u 2 ’(r 2 ) + u 2 ’(r 3 )+ … + u 1 ’(r m ) X… XX…X r 1 r 2 r 3 … r m agent 1 agent 2 sw(A opt ) =u 1 (  )+u 2 (  )+ u 2 ’(r 1 ) + u 1 ’(r 2 ) + u 2 ’(r 3 )+ … + u 2 ’(r m )

9. Modularity permits 1-deal negotiation wsp (proof cont’d) Because A is suboptimal sw(A sub ) < sw(A opt ) either u 1 ’(r 1 )<u 2 ’(r 1 ) in which case moving r 1 is IR either u 2 ’(r 2 )<u 1 ’(r 2 ) in which case moving r 2 is IR either … sw(A sub ) =u 1 (  )+u 2 (  )+ u 1 ’(r 1 ) + u 2 ’(r 2 ) + u 2 ’(r 3 )+ … + u 1 ’(r m ) sw(A opt ) =u 1 (  )+u 2 (  )+ u 2 ’(r 1 ) + u 1 ’(r 2 ) + u 2 ’(r 3 )+ … + u 2 ’(r m ) sw(A sub ) =u 1 (  )+u 2 (  )+ u 1 ’(r 1 ) + u 2 ’(r 2 ) + u 2 ’(r 3 )+ … + u 1 ’(r m ) sw(A opt ) =u 1 (  )+u 2 (  )+ u 2 ’(r 1 ) + u 1 ’(r 2 ) + u 2 ’(r 3 )+ … + u 2 ’(r m )

10. Sufficiency, necessity Sufficient condition (shown in the previous slides) if u 1 …u n  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 u 1 …u n  F. »? Answer: NO !!!! Because we can find C 1 and C 2 such that:  if u 1 …u n  C 1, then 1-deal nego wsp is permitted  if u 1 …u n  C 2, then 1-deal nego wsp is permitted  1-deal nego w.s.p. is not always permitted if u 1 …u n  C 1  C 2  class F should include C 1 and C 2 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 u 1 …u n verifies a given condition » ? ANSWER: maybe, but the condition won’t be simple, and verifying may require more than poly time Conjecture : with most compact representations (k-additive, SLP), it is NP-hard to determine wether permits 1-deal nego wsp Argument: it is NP-hard to determine if there is a 1-deal sequence from A 1 to A 2 (Dunne’s theorem using SLP utilities) Maximality There is no class F  M, such that if u 1 …u n  F, then 1-deal nego w.s-p is permitted

12. Maximality Theo: There is no class F  M, such that if u 1 …u n  F, then 1-deal nego wsp is permitted Idea of the proof: Consider any utility function u 1  M We will show that M  {u 1 } does not permit… More precisely: Given u 1  M Find u 2  M, find allocation A such that A is sub-optimal There is no IR-deal getting out of A

13. Maximality proof (1/2) Let u 1  M. Then Consider the allocations in which agent 1 owns R, and r 1,r 2 are shared among both. We can build u 2  M such that …

14. Maximality proof (2/2) More precisely, u 2 is made such that for all resources r  R, u 2 (r) << 0 for all resources r  R  {r 1,r 2 }, u 2 (r) >> 0

15. 1. Modular Functions 1. def 2. properties 2. Negotiating with side-payments (wsp) 1. Sufficiency result 2. Maximality result 3. Negotiating without side-payments (w/o.sp) 1. Sufficiency result 2. 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:  u 1 = r 1 + r 4 + r 5  u 2 = r 1 + r 2 + r 3 + r 6

17. Like-it-or-not functions Let us associate to each r i 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  u 1 = r 1 +5.r 2 + 5.r 3  u 2 = -3.r 1 -3.r 2 - 2.r 3  u 3 = r 1 - 2.r 3 These are like-it-or-not functions r1r1 r2r2 r3r3 ii 155 ii -3 -2

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 u 1 …u n  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 u 1 …u n  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 »