Mechanisms with Verification for Any Finite Domain Carmine Ventre Università degli Studi di Salerno

Task Scheduling [Nisan&Ronen’99] Allocation X  cost i (X) + t i,n = t i,j Selfish Optimal Makespan: min x max i cost i (X) Verification (observe machine behavior) no VCG! J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… b1b1 bibi bmbm …… tasks machines t1t1 titi tmtm …… types Mechanism design: payments  utility = payment - cost

Verification Give the payment if the results are given “in time”  Machine i gets job j when reporting b i,j 1. t i,j  b i,j  just wait and get the payment 2. t i,j > b i,j  no payment (punish agent i)

Why Verification? Provably better approximation  No verification  No c-APX mechanism Makespan on unrelated machines [Nisan&Ronen’99] Weighted sum on related machines [Archer&Tardos’01]  Verification  Exact mechanisms Makespan on unrelated machines [Nisan&Ronen’99] Comparable Types [Auletta et al. ‘06]  Verification  (1+  )-APX mechanism Makespan on unrelated machines [Nisan&Ronen’99] Weighted sum on related machines [Auletta et al.’06] Things become simpler  Can “recycle” existing algorithms [Auletta et al.’06] Even for two machines and exponential running time Polynomial time New lower bounds [Mu ’ Alem&Shapira ’ 06] [Christodoulou&Koutsoupias&Vidali06]

Setup Agent i holds a resource of type t i X1,…, Xk feasible solutions (how we use resources) cost i (X) = t i (X) = time utility = payment – cost Goal: minimize m(X, t ) No payment if t i (X) > b i (X) (verification) Truthful mechanism running an optimal algorithm (t 1,…,t n )

Our Contribution Can implement the optimum “in general”  Minimize any m(X,t)=m(t 1 (X),…,t n (X)) non decreasing in the agents’ costs t i (X) Can implement any optimum “in general” for compound agents  Agents declaring more than a “value” (e.g., agent controlling more than one machine) “Impossibility” results on mechanisms with verification for infinite domains

Existence of the Payments Truthfulness (single player): P(a) - a(A(a))  P(b) - a(A(b)) ab truth-telling P(b) - b(A(b))  P(a) - b(A(a)) X=A(a) Y=A(b) a(Y) - a(X) b(X) - b(Y) Must be non-negative  (a,b)  (b,a) P(a) +  (a,b)  P(b) P(b) +  (b,a)  P(a) A(  )  A( , b -i ) P(  )  P( , b -i ) Algorithm

Existence of the Payments Truthful mechanism (A, P) Can satisfy all P(a) +  (a,b)  P(b) There is no cycle of negative length abkc … [Malkhov&Vohra’04][MV’05][Saks&Yu’05] [Bikhchandani&Chatterji&Lavi&Mu'alem&Nisan&Sen’06]……

Why Verification Helps ab X a(Y) - a(X) Some edges may “disappear” Y True type is “a” but report “b”: 1.a(Y)  b(Y)  can “simulate b” and get P(b) 2.a(Y) > b(Y)  no payment (verification helps) P(a) - a(X)  P(b) - a(Y) P(a) - a(X)  - a(Y)  0 voluntary participation  0 nonnegative costs a(Y) > b(Y)

Why Verification Helps ab X a(Y) - a(X) Only these edges remain: Y a(Y)  b(Y) Negative cycles may desappear

Optimal Mechanisms Algorithm OPT: Fix lexicographic order X1  X2  …  Xk Return the lexicographically minimal Xj minimizing m(b,Xj)

Optimal Mechanisms ab XY a(Y)  b(Y) m(a(X),b -i (X))  m(a(Y),b -i (Y)) c Z b(Z)  c(Z) X is OPT(a,b -i ) c(X)  a(X) m(,b -i (Y)) is non-decreasing  m(b(Z),b -i (Z))  m(c(Z),b -i (Z))  m(b(Y),b -i (Y))  m(c(X),b -i (X))  m(a(X),b -i (X))

Optimal Mechanisms ab XY a(Y)  b(Y) m(a(X),b -i (X)) = m(a(Y),b -i (Y)) c Z b(Z)  c(Z) c(X)  a(X) = m(b(Z),b -i (Z)) = m(c(Z),b -i (Z)) = m(b(Y),b -i (Y)) = m(c(X),b -i (X)) = m(a(X),b -i (X))  Z  XX  Y X=Y=Z

Finite Domains Theorem: Truthful OPT mechanism with verification for any finite domain and any m(X,b)=m(b 1 (X),…,b m (X)) non decreasing in the agents’ costs b i (X) All vertices in a cycle lead to the same outcome Different proof of existence of exact truthful mechanism w/ verification for makespan on unrelated machines [Nisan&Ronen‘99]

(In-)Finite Domains? Nodes=declarations All vertices in a cycle lead to the same outcome Y … Nodes=outcomes X Y P(X) +  (a,b)  P(Y) D(X,Y) P(X) + D(X,Y)  P(Y) D(X,Y) = sup {  (a,b)| (a,b) edge from “X” to “Y”} P(X) P(Y) P(X) P(Y) X X D(Y,X)

(In-)Finite Domains? m(i,j) = max(i,j), two outcomes X and Y a(Y)  b(Y) abc b(X)  c(X) Y X Y X Y X b -i a(Y) - a(X) b(X) - a(Y) agent i YY X P(a) > P(c) + 7 XY -8 1

(In-)Finite Domains? SCFs implementable without verification SCFs implementable with verification There exists a class of social choice functions (SCFs) s.t. … … using the allocation graph Looking for alternative techniques

Compound Agents J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… agent 1 agent l agent k … … t1t1 titi tmtm …… types b1b1 bibi bmbm …… Each agent declares more than a type

Verification for Compound Agents Punish agent i whenever uncovered lying over one of its dimensions (e.g., machines) Collusion-Resistant mechanisms w/ verification w.r.t. known coalitions a X a(Y) - a(X) b Y a = (a 1, a 2 ) b = (b 1, b 2 ) Edge ( a, b ) exists iff a 1 (Y)  b 1 (Y) and a 2 (Y)  b 2 (Y) OPT is implementable w/verification

Compound Agents Collusion-Resistant for known coalitions mechanisms w/ verification for  makespan on unrelated machines  makespan on related machines J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… agent 1 agent l agent k … … b1b1 bibi bmbm …… Polynomial time c (1+  ) - APX Exponential time Exact mechanisms

Conclusions & Further Research OPT is “always” implementable w/ verification for finite domains  Breaking lower bounds for classical mechanisms [Archer&Tardos‘01][Bilò&Gualà&Proietti’06][NR‘99] Infinite domains and verification? Are collusion-resistant (for unknown coalitions) mechanisms w/ verification possible? Some answers in [Penna&V, Submitted]