Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions Paolo Penna and Carmine Ventre

Already on these screens... Concept of mechanisms with verification Construction of optimal mechanisms w/ verification  A class of social choice functions admitting CRMs w/ verification for any bounded domain Construction of optimal truthful mechanisms w/ verification for any bounded domain and any cost function of a certain form  Shown the technique only for finite domains collusion-resistant

Routing in Networks s Internet Change over time (link load) Private Cost No Input Knowledge Selfishness

Mechanisms: Dealing w/ Selfishness Augment an algorithm with a payment function The payment function should incentive in telling the truth Design a truthful mechanism s

Truthful Mechanisms M = (A, P) s Utility (true,,...., ) ≥ Utility (bid,,...., ) for all true, bid, and,..., M truthful if: Utility = Payment – cost = – true

Optimization & Truthful Mechanisms Objectives in contrast  Many lower bounds (even for two players and exponential running time mechanisms) Variants of the SPT [Gualà & Proietti, 06] Minimizing weighted sum scheduling [Archer & Tardos, 01] Scheduling Unrelated Machines [Nisan & Ronen, 99], [Christodoulou & Koutsoupias & Vidali 07], … Workload minimization in interdomain routing [Mu’alem & Schapira, 07], [Gamzu, 07]  & a brand new computational lower bound CPPP [Papadimitriou & Schapira & Singer, 08] Study of optimal truthful mechanisms

Collusion-Resistant Mechanisms CRMs are “impossible” to achieve  Posted price [Goldberg & Hartline, 05]  Fixed output [Schummer, 02] Unbounded apx ratios Coalition C + – ∑ Utility (true, true,,...., ) ≥ ∑ Utility (bid, bid,,...., ) for all true, bid, C and,..., in C

Describing Real World: Collusions “Accused of bribery”  1,030,000 results on Google  1,635 results on Google news Can we design CRMs using real-world information?

Describing Real World: Verification TCP datagram starts at time t  Expected delivery is time t + 1…  … but true delivery time is t + 3 It is possible to partially verify declarations by observing delivery time Other examples:  Distance  Amount of traffic  Routes availability 31 TCP IDEA ([Nisan & Ronen, 99]): No payment for agents caught by verification

(The general) Verification Setting Give the payment if the results are given “in time”  Agent is selected when reporting bid 1. true  bid  just wait and get the payment 2. true > bid  no payment (punish agent ) Utility = Payment – cost = – true

Comparison with [NR99] verification setting Two different declarations  Type  Execution time verification (reported exe time ≥ true one) Allocation depends on Payments depend on,

[NR99] verification: agent not caught (i.e., bid ≥ cost) “Physical” assumption e.g., not usable for TCP example cost = 10 mins bid = 3 hrs, Utility = Payment – reported cost = – bid NR

[NR99] verification: agent caught (i.e., bid < cost), Utility = Payment – true cost = – cost NR Mechanism truthful (resp. CR) in our verification model Mechanism truthful (resp. CR) in [NR99] verification model (Easy) Thm.

CRMs w/verification for single- parameter bounded domains Agents aka as “binary” (in/out outcomes)  e.g., controls edges any number between two known constants bid min & bid max

CRMs w/verification for single- parameter bounded domains: ideas Sufficient Properties  Pay all agents(!!!)  Algorithm 2-resistant s e e’ Truthfulness e’ has no way to enter the solution by unilaterally lying In coalition they can make the cut really expensive Utility C (true)= P e – 2 true 10+P e true 11+P e true P e’ = 0 Utility C (bid)=P e’ – 10 bid ≥ 10 + P e – 10 > Utility C (true) true

Truthful Mechanisms w/ Verification: the threshold bid < in bid > out bid A(bid, ) (A,P) truthful with verification [Auletta&De Prisco&Penna&Persiano,04] ths in out ths

2-resistant Algorithms t=(true, true,,...., ) ths b’ ths t’ ≥ b’ = b=(bid, bid,,...., ) t’ = in out ths b’ ths t’ b - =(bid,,...., ) t - =(true,,...., ) bid ≥ true(Verification doesn’t work)

Exploiting Verification: CRMs w/verification At least one agent is caught by verification Usage of the constant h for bounded domains Payment (b) = h - if out ths b’ hif in Thm. Algorithm A 2-resistant (A,Payment) is a CRM w/ verification Proof Idea.

Proof (continued) in out ths b’ ths t’ No agent is caught by verification Each is not worse by truthtelling bt in out Utility (t) == Utility (b)h - true true Utility (t) = h - ≥ h - true ths t’ = Utility (b) Payment (b) = h - if out h if in ths b’ h - ≥ h - ths t’ ths b’ h - true ≥ h - ths b’ true

Simplifying Resistance Condition t=(true, true,,...., ) ths b’ ths t’ ≥ b’ = b=(bid, bid,,...., ) t’ = in out ths b’ ths t’ b - =(bid,,...., ) t - =(true,,...., ) bid ≥ true(Verification doesn’t work) b=(bid,,...., ) t=(true,,...., ) bid ≥ true b’ =b-b- t’ =t-t- in out ths b’ ths t’ Thm. Optimal threshold-monotone algorithms with fixed tie breaking are n-resistant Optimal CRMs

Applications Optimal CRMs for:  MST  k-items auctions  Cheaper payments wrt mechanisms of previous “episode” Optimal truthful mechanisms for multidimensional agents bidding from bounded domains and non-decreasing cost functions of the form Cost(bid,..., bid )

Multidimensional Agents Outcomes = {X1,..., Xm} bid =(bid(X1),....,bid(Xm)) b=(bid,..., bid ) B(b) optimal algorithm with fixed tie breaking rule A(bid ) m optimal single- player functions View bid as a virtual coalition C of m single-parameter agents P (b) = ∑ payment (bid ) in C Lemma. If every A is m-resistant then (B,P) is truthful Thm. For non-decreasing cost function of the form Cost(bid,..., bid ) every A is threshold-monotone Every A is m-resistant (B,P) is truthful

Conclusions Optimal CRMs with verification for single- parameter bounded domains Optimal truthful mechanisms for multidimensional bounded domains  Construction tight (removing any of the hypothesis we get an impossibility result) Overcome many impossibility results by using a real-world hypothesis (verification) For finite domains: Mechanisms polytime if algorithm is Can we deal with unbounded domains? Threshold-monotone vs. utilitarian algorithms