1 Foundations of Distributed Algorithmic Mechanism Design Joan Feigenbaum Yale University

2 Two Views of Multi-agent Systems CSECON Focus is on Computational & Communication Efficiency Agents are Obedient, Faulty, or Byzantine Focus is on Incentives Agents are Strategic

3 Secure, Multiparty Function Evaluation... t 1t 1 t 2t 2 t 3t 3 t n-1 t nt n O = O (t 1, …, t n ) Each i learns O. No i can learn anything about t j (except what he can infer from t i and O ).

4 SMFE Literature Agents 1, …, n are obedient or byzantine. Obedient agents limited to probabilistic polynomial time. (Sometimes, byzantine agents too.) Typical assumption: at most n/3 byzantine agents Typical successful protocol: − All obedient agents learn O. − No byzantine agent learns O or t j for an obedient j.

5 Internet Computation Both incentives and computational and communication efficiency matter. “Ownership, operation, and use by numerous independent self-interested parties give the Internet the characteristics of an economy as well as those of a computer.”  DAMD: “Distributed Algorithmic Mechanism Design”

6 Outline DAMD definitions and notation Short example: Multicast cost sharing General open questions Long example: Interdomain routing

7 Definitions and Notation t 1t 1 t nt n Agent 1 Agent n Mechanism... p 1p 1 a 1a 1 p np n a na n O (Private) types: t 1, …, t n Strategies: a 1, …, a n Payments: p i = p i (a 1, …, a n ) Output: O = O (a 1, …, a n ) Valuations: v i = v i (t i, O ) Utilities: u i = v i + p i (Choose a i to maximize u i.)

8 “Strategyproof” Mechanism For all i, t i, a i, and a –i = (a 1, …, a i-1, a i+i, …, a n ) v i (t i, O (a –i, t i )) + p i (a –i, t i )  v i (t i, O (a –i, a i )) + p i (a –i, a i ) “Truthfulness” “Dominant Strategy Solution Concept” Nisan-Ronen ’99: Polynomial time O ( ) and p i ( )

9 Example: Task Allocation Input: Tasks z 1, …, z k Agent i ’s type: t = (t 1, …, t k ) (t j is the minimum time in which i can complete z j ) Feasible outputs: Z = Z 1 Z 2 … Z n (Z i is the set of tasks assigned to agent i) Valuations: v i (t, Z) = −  t j Goal: Minimize max  t j i i i i i i zj  Z izj  Z i zj  Z izj  Z i iZ i

10 Nisan-Ronen Min-Work Mechanism O (a 1, …, a n ): Assign z j to agent with smallest a j p i (a 1, …, a n ) =  min a j [NR99]: Strategyproof, n-Approximation Open Questions: Average case(s)? Distributed algorithm for Min-Work? i i’ i  i’ zj  Z izj  Z i

11 Distributed AMD [FPS ’00] Agents 1, …, n Interconnection network T Numerical input {c 1, … c m }  O ( |T | ) messages total  O (1) messages per link  Polynomial time local computation  Maximum message size is polylog (n, |T |) and poly (  ||c j || ). “Good network complexity” j=1 m

12 Multicast Cost Sharing Mechanism-Design Problem ,23,0 1,26,710 Users’ types Link costs Source Which users receive the multicast? Receiver Set Cost Shares How much does each receiver pay?

13 Two Natural Mechanisms  Group-strategyproof  Budget-balanced  Minimum worst-case efficiency loss  Bad network complexity  Strategyproof  Efficient  Good network complexity  Shapley value  Marginal cost

14 Marginal Cost Receiver set: R* = arg max NW(R) R NW(R)   t i – C(T(R)) i  Ri  R Cost shares: p i = 0 if i  R* t i – [ NW ( R*( t ) ) – NW ( R* ( t | i 0 ) ) ] o.w. { Computable with two (short) messages per link and two (simple) calculations per node. [FPS ’00]

15 Shapley Value Cost shares: c( l ) is shared equally by all receivers downstream of l. (Non-receivers pay 0.) Receiver set: Biggest R* such that t i  p i, for all i  R* Any distributed algorithm that computes it must send  (n) bits over  (|T |) links in the worst case. [FKSS ’02]

16 Group-Strategyproof, BB Cost Sharing “Representative Hard Problem” to solve on the Internet Hard if it must be solved: By a distributed algorithm Computationally efficiently Incentive compatibly Becomes easy if one requirement is dropped. Open Question: Find other representative hard problems.

17 Open Question: More (Realistic) Distributed Algorithmic Mechanisms  Caching  P2P file sharing  Interdomain Routing  Distributed Task Allocation  Overlay Networks

18 Open Question: Strategic Modeling In each DAMD problem, which agents are  Obedient  Strategic [  Byzantine ] [  Faulty ] ?

19 Open Question: What about “provably hard” DAMD problems? AMD approximation is subtle; can destroy strategyproofness “Feasibly dominant strategies” [NR ’01] “Strategically faithful” approximation [FKSS ’01] “Tolerable manipulability” [FKSS ’01]

20 Revelation Principle If there is a mechanism ( O, p) that implements a design goal, then there is one that does so truthfully.... Agent 1 Agent n FOR i  1 to n SIMULATE i to COMPUTE a i O  O (a 1, …, a n ); p  p (a 1, …, a n ) p 1p 1 t 1t 1 p np n t nt n O Note: Loss of privacy Shift of computational load

21 Open Question: Design Privacy-Preserving DAMs Cannot simply “compose” a DAM with a standard SMFE protocol. −   (n) obedient agents − Unacceptable network complexity − Agents don’t “know” each other. New SMFE techniques?

22 Open Question: Can Indirect Mechanisms be More Efficient w.r.t Computation or Communication? Mechanism computation Agent computation Communication

23 Interdomain-Routing Mechanism-design Problem Inputs: Transit costs Outputs: Routes, Payments Qwest Sprint UUNET WorldNet Agents: Transit ASs

24 Problem Statement Strategyproofness “BGP-based” distributed algorithm Lowest-cost paths (LCPs) Per-packet costs {c k } Agents’ types: {route(i, j)}Outputs: (Unknown) global parameter: Traffic matrix [T ij ] {pk}{pk} Payments: Objectives:

25 Previous Work Nisan-Ronen, 1999 Single (source, destination) pair Links are the strategic agents “Private type” of l is c l (Centralized) strategyproof, polynomial-time mechanism Hershberger-Suri, 2001 p l  d G|c l =  - d G|c l =0 Compute m payments as quickly as 1

26 Our Formulation vs. NR, HS Nodes, not links, are the strategic agents. All (source, destination) pairs Distributed “BGP-based” algorithm More realistic model Advantages: Deployable via small changes to BGP

27 Notation LCPs described by indicator function: 1 if k is on the LCP from i to j, when cost vector is c 0 otherwise c Ι   (c 1, c 2, …, , …, c n ) { I k (c; i,j)  k

28 A Unique VCG Mechanism For a biconnected network, if LCP routes are always chosen, there is a unique strategyproof mechanism that gives no payment to nodes that carry no transit traffic. The payments are of the form p k =  T ij, where = c k I k (c; i, j) + [  I r (c Ι  ; i, j ) c r -  I r (c; i, j ) c r ] Theorem 1: p ij k rr i,j p ij k k

29 Features of this Mechanism Payments have a very simple dependence on traffic T ij : payment p k is the sum of per-packet prices. Price is 0 if k is not on LCP between i, j. Cost c k is independent of i and j, but price depends on i and j. Price is determined by cost of min-cost path from i to j not passing through k (min-cost “k-avoiding” path). p ij k k k k

30 BGP-based Computational Model (1) Follow abstract BGP model of Griffin and Wilfong: Network is a graph with nodes corresponding to ASes and bidirectional links; intradomain-routing issues are ignored. Each AS has a routing table with LCPs to all other nodes: Entire paths are stored, not just next hop. Dest. LCP LCP cost AS3AS53AS1 AS7AS22

31 BGP-based Computational Model (2) An AS “advertises” its routes to its neighbors in the AS graph, whenever its routing table changes. The computation of a single node is an infinite sequence of stages: Receive routes from neighbors Update routing table Advertise modified routes Complexity measures: − Number of stages required for convergence − Total communication

32 Towards Distributed Price Computation = c k + Cost ( P -k (c; i,j) ) – Cost ( P(c; i,j) ) LCPs to destination j form a tree Use data from i ’s neighbors a,b,d to compute tree edge non-tree edge j a bd i. p ij k k

33 Constructing k-avoiding Paths Three possible cases for P -k (c; i, j): j a b d i k i ’s neighbor on the path is (a) parent (b) child (d) unrelated In each case, a relation to neighbor’s LCP or price, e.g., (b) = + c b + c i is the minimum of these values. p ij k p bj k p ij k

34 A “BGP-based” Algorithm AS3AS5 c(i,1) AS1 c1c1 Dest.cost LCP and path prices LCP cost AS1 LCPs are computed and advertised to neighbors. Initially, all prices are set to . Each node repeats: − Receive LCP costs and path prices from neighbors. − Recompute path prices. − Advertise changed prices to neighbors. Final state: Node i has accurate values. p ij k p i1 3 5

35 Performance of Algorithm d’ = max i,j,k || P -k ( c; i, j ) || d = max i,j || P ( c; i, j ) || Our algorithm computes the VCG prices correctly, uses routing tables of size O(nd) (a constant factor increase over BGP), and converges in at most (d + d’) stages (worst-case additive penalty of d’ stages over the BGP convergence time). Theorem 2:

36 Open Question: Strategy in Computation Mechanism is strategyproof : ASes have no incentive to lie about c k ’s. However, payments are computed by the strategic agents themselves. How do we reconcile the strategic model with the computational model? “Quick fix” : Digital Signatures [Mitchell, Sami, Talwar, Teague] Is there a way to do this without a PKI?

37 Open Question: Overcharging In the worst case, path price can be arbitrarily higher than path cost [Archer&Tardos, 2002]. This is a general problem wiith VCG mechanisms. Statistics from a real AS graph, with unit costs: Mean node price : 1.44 Maximum node price: 9 90% of prices were 1 or 2 How do VCG prices interact with AS-graph formation? Overcharging is not a major problem!