1. Price of Anarchy, and Strategyproof Network Protocol Design Xiang-Yang Li Department of Computer Science Illinois Institute of Technology Collaborated with: Weizhao Wang, Zheng Sun

2. Traditional Algorithms, Protocols Efficiency Time, space, communication efficiency Assumption Participants act as instructed Not always true Faulty ones  Fault-tolerant computing Malicious ones  Security, and Trusted computing Selfish ones  Strategyproof computing

3. Preliminaries and Related Works Non-cooperative games Price of Anarchy Strategyproof mechanisms for routing Unicast Multicast Cooperative games Cost and payment sharing Conclusion and future work Outline

4. Example: wireless network routing, need nodes to relay packets, but Nodes are battery powered Nodes are selfish (self-incentive) Denying/lying can result disaster for system Example: TCP/IP congestion control Additive increase, Multiplicative decrease Terminals can deviate from this and benefit Why need truthful computing?

5. How to deal with selfish nodes? Reputation based methods Routing through nodes with good reputation Nodes are rated by peers Pay each node its declared cost Node will manipulate its declared “cost” to increase its profit May reach a stable point: no node will unilaterally change its declared cost---Nash Pay each node some payment Node maximizes its profit when it reports cost truthfully So relieve nodes from manipulating declared cost

6. Models Non-cooperative games Extensive (Sequential) Game– Chess Strategic (Simultaneous) Game ( ) – Scissor-paper-stone Topics to discuss Price of anarchy Strategyproof mechanism design (assume no collusion) Cooperative games Transferable payoffs (side payments) Non-transferable payoffss Topics to discuss Sharing the cost of providing service Sharing the payments to selfish service providers

7. Non-cooperative Games

8. Strategic Game It composes of A set of n players (or called agents) For each player, a set of strategies For each player, a payoff function that gives payoff to him based on n actions chosen by n players Selfish player Chooses best action to maximize its payoff, given others’ actions Not necessarily the “best” if cooperate

9. Example: Prisoners Dilemma 1 2cooperatedefect cooperate(R,R)(S,T) defect(T,S)(P,P) Both players view: T>R > P>S e.g., T=10, R=8, P=4, S=1 Nash equilibrium (defect,defect)  (p,p) Global optimum (R,R) if 2R>T+S

10. Price of anarchy What if we let the selfish agents play out with each other? Stable point (Nash equilibrium): no agent can unilaterally switch its strategy to improve its payoff Not always exist Conjecture: NP-hard to find one The performance at stable point Worst ratio of this over the optimum if cooperated --- price of anarchy E.g., price of anarchy of prisoner’s dilemma is R/P Price of anarchy of a protocol could be large (as routing)

11. Algorithm Mechanism Design Instead of letting players play out, design incentives to influence the actions Knows what selfish players will do under incentives The system performance is ensured Typical incentives Monetary values Differentiated services, and so on

12. Algorithm Mechanism Design N players Private type t i Strategy from A i Mechanism M=(O,P) Output O(a) Payment p(a) Player i Valuation: Utility:  N wireless nodes  Private cost c i  Strategies: all possible costs  Mechanism  Output O(c): a path  Payment p(c)  Node i  Valuation:  Utility: Unicast game

13. Algorithm Mechanism Design Truthful (Strategyproof) Mechanism Design Incentive Compatibility: for every agent i, revealing its true type is a best strategy regardless of whatever others do (dominant strategy). Individual Rationality: Every agent must have non-negative utility if reveals its true private input. Other Desirable Property Polynomial time complexity Output Payment

14. Example: Network Protocols Unicast Truthful payment scheme (VCG scheme) Our contribution: Fast Computation Collusion Among nodes: Negative results Multicast VCG not applicable Several truthful mechanisms for structures: LCPS, VMST, PMST, Steiner Tree. Payment Computing, and sharing

15. Unicast 8 6 7 7 9 5 1 7 Node v k costs c k to relay (private knowledge) Each node v k reports a cost d k Find the shortest path from node v 0 to node v 9 based on reported costs d Compute a payment p k for node v k based on d  Objective: Find a payment p k (d) so node maximizes utility when d k =c k

16. Strategyproof Unicast Scheme Output Least cost path from s to t, by LCP(s, t, G) Payment to a relay node v k Remove it and its incident links Compute the shortest path from s to t The payment to v k is Otherwise the payment is 0

17. Unicast Mechanism A VCG mechanism Output maximizes the total valuations Payment is VCG family Distributed Computing By Feigenbaum, Papadimitriou, Sami, Shenker But still lots to be solved It is the selfish agents who do the computing!

18. VCG Mechanism Who designed? Vickrey(1961); Groves(1973); Clarke(1971) What is VCG Mechanism? A VCG Mechanism is truthful.

19. 5 5 8 6 7 7 9 7 7 Total payment is 8+9+10 =27 instead of actual cost 18. Overpayment ratio: Unicast Payment Calculation

20. Overpayment- Frugality Ratio VCG mechanism Overpayment could be arbitrarily large LCP+ k (LCP2-LCP), k is hop of LCP Any truthful mechanism for unicast Overpayment could be arbitrarily large LCP+ min(k1,k2) (LCP2-LCP)/2, k1 is hop of LCP, k2 is hop of LCP2 Proved by Elkind et al, 2004

21. Payment calculation for one node Dijstra’s algorithm Time complexity O(n log n+m) Payment calculation for all nodes on the LCP Using Dijstra’s algorithm for every node Time complexity O(n 2 log n+nm) We can calculate it faster! Our Result: Payment calculation for all nodes on the LCP could be done in O(n log n+m) which is optimal. Fast Payment Calculation

22. Problem Statement: A graph, a cost vector for all nodes or links, k receiving nodes R. The cost is private value Find a spanning tree to minimize Multicasting 3 4 7 5 9 2 1

23. Fundamental differences For unicast LCP (max total valuations) can be found efficiently For multicast NP hard to find min-cost tree (max total valuations) with only ln n approximation for node weighted graph and O(1) for link weighted graph. This difference leads to VCG does not apply for multicast How to design truthful mechanisms? Generally, replacing optimum with approximation leads to non-truthfulness (Nisan, Ronen)

24. Challenges Given a method A constructing a multicast structure, can we design a strategyproof mechanism M=(A,P) using it as output? Necessary and sufficient conditions If exists, how to? Efficient computing of payment Fair sharing of payment

25. Necessary and Sufficient Conditions A multicast method implies a strategyproof mechanism iff It is monotone: still selects a relay node if it has a less cost Monotone structures Least Cost Path Tree (LCPT) Based Virtual Minimum Spanning Tree Based Steiner Tree Based

26. Strategyproof Payment Scheme Define Truthful Payment Schemes Network is bi-connected (avoid monopoly) Payment to a relay node Fix others’ costs, the maximum cost it could declare while still in the structure selected Not selected cost

27. Payment Is Truthful! Individual Rationality (IR): non-negative utility Incentive Compatibility (IC) A node lies up its cost to Originally it is a relay node Originally it is not a relay node A nodes lies down its cost to Originally it is a relay node Originally it is not a relay node

28. Payment Optimality The payment scheme is optimal regarding every individual payment among all truthful payment scheme based on this structure

29. Computing Payment Threshold defines a payment Question left: how to find the threshold efficiently? Illustrate for structure Least Cost Path Tree (LCPT) Based

30. Structure (node or link or both) Calculate all shortest paths from source node to receivers Combine these shortest paths The structure is a tree called Least Cost Path Tree (LCPT) Payment Scheme Calculate the payment for node v k based on every LCP containing v k Choosing the maximum of these payments as the final payment 3 4 7 3 9 2 1 LCPT Based Method

31. Other Structures Virtual Minimum Spanning tree Construct the virtual complete graph K(G) Nodes are receivers, plus source node Edges are LCP between two end-points Find the MST on K(G), say V MST(G) All agents on VMST(G) are selected General link weighted Steiner Tree NP-Hard, constant approximation methods exist Efficient computing of payments General Node weighted Steiner Tree NP-Hard, best approximation ratio O(ln k) Efficient computing of payments See our MobiCom 2004 paper for more details

32. Cooperative Games

33. What is cooperative game A set of agents N perform some task together and get a value v(N) how to share the value among them The sharing should be fair! Does share encourage cooperation? More member, larger shared value

34. Example: Cost Sharing Given a set of players N The cost of C(S) for every is known The cost is cohesive: C(S+T)<= C(S)+C(T) Cost allocation Share the cost among players Budget balance Be fair– core: Cost sharing scheme Share the cost C(S) for every S: Cross-monotone

35. Multicast Cost Sharing(fixed tree) Given a structure for multicast The cost of each relay agent is known A fixed tree from the source to all receivers Share the cost among receivers Budget balance Be fair– core Cross-monotone Methods: Shapley Value 3 4 7 3 9 2 1

36. Equally share for downstream receivers 3 4 7 3 9 2 1 4 3 3/2+3/2=3

37. Multicast Cost Sharing-Valuation Given a structure for multicast The cost of each relay agent is known A fixed tree from the source to all receivers Share the cost among receivers Budget balance Be fair– core Cross-monotone Each receiver has a valuation, and participates only if So need select subset of receivers

38. With valuation 3 4 7 3 9 2 1 5 2 5 3 4 7 3 9 2 1 4 5 2 5 4 3 3 3 4 7 3 9 2 1 4 5 2 5 6

39. Cost Sharing (no fixed tree) All receivers must get the data Find an efficient tree Share the cost of tree among receivers fairly? Various concepts of fair: core, bargaining set, etc  -Core:  -Budget balance “fair” Tight bound No allocation can recover more than fraction of optimum cost Conjecture: Exist an allocation can recover fraction of cost

40. Cost Sharing (no fixed tree) Cross monotonic  -Core:  -Budget balance “fair” Cross monotone Tight bound No allocation can recover more than fraction of optimum cost of Shapley value on LCPT can recover fraction of cost and also the actual cost!

41. Sharing Payment Since the relay agents may be selfish, we need share the payments to relay agents among receivers Need to be fair and encourage cooperation No free rider: sharing is at least some factor of the payment needed if it acts alone Cross-monotonic: more population, less sharing No negative transfer Budget balance

42. Sharing payment: LCPT payment Mechanism using LCPT as output 8 6 7 9 4 4 14 15 5 14 7/2+(15-8)=11.5 7/2+5=8.5

43. Properties No negative transfer Budget balance Cross-monotonic No-free rider Dummy: sharing is its cost if marginal payment = payment of unicast Symmetry: shared payments are same if two are interchangeable

44. Summary Computing in selfish environment Non-Cooperative Games Price of anarchy Strategyproof mechanism Cooperative Games Cost sharing Payment sharing How to share the payment for other structures

45. Questions and Comments

46. Game Theory, studied in Neoclassical economics Mathematics Other social and behavioral sciences (Psychology) Computer Science Game Theory History John von Neumann, Oskar Morgenstern (1944) “Theory of games and economics behavior” Prisoner's Dilemma (1950) John Nash: Non-cooperative game; Nash equilibrium (1951) Selfish via Game Theory