Algorithms for Selfish Agents Carmine Ventre Università degli Studi di Salerno

Algorithms for Selfish Agents or… … having some “scientific” argument to use for the movie “A Beautiful Mind”  … being more confident with Nash’s madness! … merging research of different fields

Research Classification Economics Computer Science “Worst-case equilibria” by E. Koutsoupias, C. H. Papadimitriou in STACS ‘99

Auctions First price sealed bid auction 6 Problems?It is not truthful (e.g., auctioneer can not maximize his own revenue) A B

Vickrey Auctions 10 Second price sealed bid auction Bid 8 11 Utility is 0 in place of 1 (= 10 – 9) Bid 12 Utility is -1 (= 10 – 11) in place of 0 9 This is truthful! A B

Vickrey Auctions Features Incentives to bid truthfully  Utility is maximized by reporting the true valuation Extend to more goods case  Winners pay the first non-winning bid (Not common, however…) Used in practice  former Czechoslovakia to refinance credit  Guinea, Nigeria, and Uganda for foreign exchange  eBay and Google use mild variations of Vickrey auction  Stamp collecting market Generalize to the concept of mechanism

Mechanisms Augment an algorithm with a payment function  i.e., design a truthful mechanism The payment function should incentive in telling the truth s

VCG Mechanisms s M = (A, P) P e = b e if e is selected (0 otherwise) 9 Utility(3) = payment(3) – cost(3) = 3 – 3 = 0 Utility(9) = payment(9) – cost(9) = 9 – 3 = 6 valuation P e = A e=∞ – A e=0 if e is selected (0 otherwise) M is truthful iff A is optimal Algorithmic mechanism design by N. Nisan and A. Ronen in STOC ’99 (GEB ‘01) A e=0 = A e – b e

Vickrey Auction (& VCG Mechanism) Weakness It works only for utilitarian problems: i.e., maximizes the social welfare (e.g., it does not maximize seller revenue)  Adaptation to non-utilitarian problems  Verification Model It is not budget balanced  Cost-Sharing Budget Balance Mechanisms It is vulnerable to collusion  Cost-Sharing Budget Balance Mechanisms  Verification model VCG mechanisms require computation of optimal solutions  Computational issues! … Utilitarian problems: objective is to maximize the social welfare (  i valuation i (X)) BB mechanisms: sum of payments equals the cost of the solution

Cost-Sharing Mechanisms

Why Cost-Sharing Methods? Town A needs a water distribution system  A’s cost is € 11 millions Town B needs a water distribution system  B’s cost is € 7 millions A and B construct a unique water distribution system for both cities  The total cost is € 15 millions Why not collaborate and save € 3 millions? How to share the cost? Town A Town B

Multicast and Cost-Sharing A service provider s Selfish customers U Who is getting the service? How to share the cost? real worth is 7 is worth 5 (  7) PiPi Accept or reject the service?

Selfish Agents Each customer/agent  has a private valuation v i for the service  declares a (potentially different) valuation b i  pays P i for the service Agents’ goal is to maximize their own utility: u i (b i ) := v i – P i (b i ) Accept iff my utility ¸ 0!

Coping with Selfishness: Mechanism Design Algorithm A  Who gets serviced (Q(b))  How to reach Q(b) (Construct tree T) Payment P  How much each user pay M = (A, P) bibi bjbj P1P1 P4P4 P3P3 P2P2

M’s Truthfulness (or Strategyproofness) For all others players’ declarations b -i it holds u i = u i (v i, b -i ) ¸ u i (b i, b -i ) = u i for all b i (ie, truthtelling is a dominant strategy) M = (A, P) vivi

M’s Group Strategyproofness U Coalition C No one gains At least one looses (ie, u i < u i ) C is useless Breaks off C Does this definition fit our intuition of collusion-resistant mechanisms?

Mechanism’s Requirements Budget Balance (BB)   i  T P i (b) = COST(T) Efficiency (NW): maximize NET WORTH(T) := WORTH(T) - COST(T) where WORTH(T):=  i  T v i … (natural requirements) BB and efficiency are mutually exclusive! Efficiency  No Group strategy-proof

Efficient Mechanisms for Multicast Transmissions in Wireless Networks [Penna & V, SIROCCO ‘04]

Wireless transmission Power(i)= d(i,j) α = range(i) α, α>1 (empty space α = 2) A message sent by station i to j can be also received by every station in transmission range of i Modeling wireless transmission network as a graph  Network communication graph i j d(i,j) α

Wireless multicast transmission Who receives Napoli-Juventus How to transmit Goal: maximize Benefit – Cost Nello 1€ 10€1€ 3€ Carmine 1€Clemente 10€Marco 30€ Pino 50€ known private source NW(T) = WORTH(T) – COST(T) Designing an efficient mechanism

VCG Trick (marginal cost mechanism) Utilitarian problem:   X  sol, measure(X)=  i valuation i (X) A opt computes X  sol maximizing measure(X)  P VCG : M=(A opt, P VCG ) is truthful

VCG Trick (marginal cost mechanism) Making our problem utilitarian: measure(X) valuation i (X) WORTH(X)-COST(X) =  i iXiX vivi = WORTH(X) vivi cici Initially, charge to every receiver i the cost c i of its ingoing connection - c i - COST(X) P i = c i + P VCG Adaptation to non-utilitarian problems

Designing (Distributed) Optimal Algorithms Tree Networks  Dynamic Programming Poly-Time algorithm  Extensions to trees with metric-free edges Graph Networks  NP-hard k s.t. c k ≤ c j i j cjcj vivi k gets c j -c k units of credit

Cost-Sharing Budget-Balance Mechanisms [Penna & V, WAOA ’04] [Penna & V, SIROCCO ’05] [Penna & V, STACS ’06] Cost-Sharing Budget Balance Mechanisms

How to build BB, GSP Mechanisms Idea: associate prices to service set U Q   (Q,i) = COST(Q) Cost-sharing methods: distribute COST(Q) among users in Q  (Q,i)  0  (Q,i) = 0, i  Q

How to build BB, GSP Mechanisms Cost-sharing method  (, )  Mechanism M(  )  (Q,i) (Q,i) U Drop i Q > b i

U Q 1 =U How to build BB, GSP Mechanisms Q3Q3 QkQk … Q2Q2 Prices do not decrease Group Strategyproof  (Qk,i) (Qk,i)  (Q2,i) (Q2,i)  (Q3,i) (Q3,i) P i =  (Q k,i) Changes … Monotonicity… [Moulin & Shenker ’97] & [PV04] Cost-sharing method  (, )  Mechanism M(  ) … for all Q subsets of U… for all Q (possibly) outputted by M Cross… Self Cross…

Self cross monotonicity: an example Q COST(Q) s 50% s Pay less than before This is not a cross monotonic cost sharing method!

Self cross monotonicity: an example (2) Q COST(Q) s 100% s Pay less than before This guy pays 0 M(  ) cannot drop him Idea: some Q µ U do not “appear”. We need  monotone only for possible subsets generated by M(  ) This is not a cross monotonic cost sharing method!

Sequential Algorithms A is sequential if for some bid vectors reaches a chain of sets Q 1, …, Q |U|, ; Sequential algorithms admits a self cross- monotonic cost-sharing method U Q 1 =U Q3Q3 Q |U| … Q2Q … Q |U|+1 = ; BB & GSP Mechanisms

Optimal Sequential Algorithm for Steiner Tree Game s prune Q MST(Q) opt Steiner tree T +  = opt T +  U s MST v is the last node added by Prim’s MST algorithm s u Q v  s u Q  v s T*T* > Q u  v pay v

Adding Fairness to Our Mechanisms Payment is still self cross-monotonic Is it possible to have no free rider?  No! Unless P=NP s prune Q MST(Q) U s MST pay opt Steiner tree

Can we do better without Sequential Algorithms? M = (A, P) M for 2 usersA is sequential “Natural” GSP MechanismsA is sequential M is SP, BB, …

Mechanisms with Verification [Ferrante, Parlato, Sorrentino & V, WAOA 2005] [Auletta, De Prisco, Penna, Persiano & V, ICALP 2006] [V, WINE 2006] [Penna & V., Submitted] Verification model

Selfish Task Scheduling M1M2M4M3M5 t1t1 t2t2 t3t3 t4t4 t5t5 b1b1 b2b2 b3b3 b4b4 b5b5 t i = 1 / s i (ie, the inverse of the speed) Optimal Makespan: min x max i t i (X) Mechanism design: payments  utility = payment - cost no VCG! Allocation X  cost = t i (X) = t i load i (X) Awarded independently from the execution!

Verifiable Selfish Agents t i = 1 i underbids 1/2 1 3 i’s release time should be 2 but… … i’s finishing time is 4 i overbids i can wait 2 time slots delivering the results in the right time IDEA ([Nisan & Ronen, 99]): No payment for underbidding agents Verification is impossible! t i (X) = load i (X) ¢ t i i bids from the set {1/2, 1, 2} Verification = observe jobs’ release time

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

The Power of Verification bibi load i Classical mechanismsMechanisms w/ Verification algorithms Payment functions NO! TRUTHFUL bibi load i NO! TRUTHFUL bibi load i titi P i (b i, b -i )= W max / b i (= W max ¢ s i ) Related to max (possible) supported cost Scaling up for general speeds Unique Not unique [Archer & Tardos, ‘01] [Auletta & al, ‘04]

The Power of Verification: Breaking Lower Bounds M1M2M4M3M5 b1b1 b2b2 b3b3 b4b4 b5b5 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 weight priority t1t1 t2t2 t3t3 t4t4 t5t5 Goal: Design a polytime truthful mechanism optimizing the maximum weighted completion time (ie, weighted sum scheduling) No 1.54-apx truthful mechanism without verification [Archer & Tardos, 2001] (1+  )-APX truthful mechanism w/ verification for a constant number of machines Efficient APX truthful mechanisms w/verification: c-APX algorithm A  c(1+  )-APX mechanism

Generalizing Verification Setting Give the payment if the results are given “in time” (ie, consistently with b i )  For the outcome computed in b i, ie, X 1. t i (X)  b i (X)  just wait and get the payment 2. t i (X) > b i (X)  no payment (punish agent i) ti(¢)ti(¢) bi(¢)bi(¢) General cost functions (e.g., router latency)

(Optimal) Mechanisms with Verifications J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… b ij b i1 b in …… agent 1 agent l agent k … … There exists truthful mechanism with verification We don’t if truthful mechanisms without verification do exist polytime Breaking lower bounds for classical mechanisms concerning many natural problems (eg, variants of SPT problem) Goal: minimizing the makespan (althougt not polynomial-time) Given an algorithm c-apx… a c(1+  )-apx an exact bibi b1b1 bmbm

Optimal Collusion-Resistant Mechanisms w/ Verification Coalition C GSP do not consider side payments U + – Collusion-Resistant mechanisms are impossible unless using posted-price ([Goldberg & Hartline, 2002]) If OPT is truthful via VCG mechanism without verification Exists a VCG-like payment function such that OPT is n-collusion-resistant with verification

Conclusions You have more insights on why Nash got mad Cost-Sharing Games  Simple techniques… … lead to polynomial-time cost-sharing mechanisms for NP- Hard problem Steiner Tree … not so unfair (unless P=NP) … characterize natural class of cost-sharing mechanisms Mechanisms with Verification  More powerful model… … breaking known lower bounds for natural problems … dealing with a strong notion of agents’ collusion

Some references Penna & V. "Sharing the cost of multicast transmissions in wireless networks" (SIROCCO 2004) Penna & V. "More Powerful and Simpler Cost-Sharing Methods (when cross monotonicity is the wrong way)". (WAOA 2004) Penna & V. "Free-riders in Steiner tree cost-sharing games". (SIROCCO 2005) Ferrante, Parlato, Sorrentino & V. "Improvements for Truthful Mechanisms with Verifiable One-Parameter Selfish Agents". (WAOA 2005) P. Penna & V. "The Algorithmic Structure of Group Strategyproof Budget- Balanced Cost-Sharing Mechanisms". (STACS 2006) Auletta, De Prisco, Penna, Persiano & V. "New Constructions of Mechanisms with Verification“. (ICALP 2006) V. "Mechanisms with Verification for Any Finite Domain". (WINE 2006) Penna & V. “Optimal Collusion-Resistant Mechanisms with Verification”. (It’s going to be) Submitted for publication. Others: Penna & V. "Energy-efficient broadcasting in ad-hoc networks: combining MSTs with shortest-path trees". (PE-WASUN 2004) V. & Visconti. "New Definitions and Results for Plaintext Aware and Completely Non-Malleable Encryption Schemes“. Submitted for publication. Wireless Energy Consumption Cryptography