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

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on maximal classes of utility functions for efficient resource at a time negotiation

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

Yann Chevaleyre, LAMSADE

University of Paris 9 - Dauphine

mara the setting
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)
problematic
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

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

u’(R)

properties of modular functions

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
modularity permits 1 deal negotiation wsp
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)
modularity permits 1 deal negotiation wsp proof
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)

modularity permits 1 deal negotiation wsp proof cont d
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)

sufficiency necessity
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!!!
necessity maximality
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
maximality
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
maximality proof 1 2
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 …
maximality proof 2 2
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
slide15
Modular Functions
    • def
    • properties
  • Negotiating with side-payments (wsp)
    • Sufficiency result
    • Maximality result
  • Negotiating without side-payments (w/o.sp)
    • Sufficiency result
    • Maximality result
sufficiency w o s p
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
like it or not functions
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
like it or not functions cont d
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).
sufficiency maximality
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
conclusion
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 »