Price of anarchy and strategyproof network protocol design
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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. Traditional Algorithms, Protocols. Efficiency Time, space, communication efficiency Assumption

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Price of Anarchy, and Strategyproof Network Protocol Design

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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


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


Outline

  • 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


Why need truthful computing?

  • 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


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


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


Non-cooperative Games


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


Example: Prisoners Dilemma

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


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)


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


  • N wireless nodes

    • Private cost ci

    • Strategies: all possible costs

  • Mechanism

    • Output O(c): a path

    • Payment p(c)

  • Node i

    • Valuation:

    • Utility:

Algorithm Mechanism Design

Unicast game

  • N players

    • Private type ti

    • Strategy from Ai

  • Mechanism M=(O,P)

    • Output O(a)

    • Payment p(a)

  • Player i

    • Valuation:

    • Utility:


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


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


Unicast

  • Node vkcosts ckto relay (private knowledge)

  • Each node vk reports a cost dk

  • Find the shortest path from node v0 to node v9 based on reported costs d

  • Compute a payment pk for node vk based on d

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  • Objective: Find a payment pk(d) so node maximizes utility when dk =ck


Strategyproof Unicast Scheme

  • Output

    • Least cost path from s to t, by LCP(s, t, G)

  • Payment to a relay node vk

    • Remove it and its incident links

    • Compute the shortest path from s to t

    • The payment to vk is

  • Otherwise the payment is 0


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!


VCG Mechanism

  • Who designed?

    • Vickrey(1961); Groves(1973); Clarke(1971)

  • What is VCG Mechanism?

  • A VCG Mechanism is truthful.


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Overpayment ratio:

Unicast Payment Calculation

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Total payment is 8+9+10 =27 instead of actual cost 18.

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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


Fast Payment Calculation

  • 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(n2 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.


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Multicasting

  • 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


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)


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


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


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

Not selected

selected

cost


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


Payment Optimality

  • The payment scheme is optimal regarding every individual payment

    • among all truthful payment scheme based on this structure


Computing Payment

  • Threshold defines a payment

  • Question left: how to find the threshold efficiently?

  • Illustrate for structure

    • Least Cost Path Tree (LCPT) Based


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LCPT Based Method

  • 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 vk based on every LCP containing vk

    • Choosing the maximum of these payments as the final payment

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Other Structures

  • Virtual Minimum Spanning tree

    • Construct the virtual complete graphK(G)

      • Nodes are receivers, plus source node

      • Edges are LCP between two end-points

    • Find the MST onK(G), say VMST(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


Cooperative Games


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


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


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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


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3/2+3/2=3

Shapley Value

  • Equally share for downstream receivers


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


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With valuation

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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


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!


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


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7/2+(15-8)=11.5

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7/2+5=8.5

Sharing payment: LCPT payment

  • Mechanism using LCPT as output


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


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


Questions and Comments


Selfish via Game Theory

  • 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)


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