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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

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
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
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
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
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
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
strategic game
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
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
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
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
algorithm mechanism design1

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 design2
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
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
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
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
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
VCG Mechanism
  • Who designed?
    • Vickrey(1961); Groves(1973); Clarke(1971)
  • What is VCG Mechanism?
  • A VCG Mechanism is truthful.
unicast payment calculation

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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
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
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.
multicasting

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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
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
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
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
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
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
Payment Optimality
  • The payment scheme is optimal regarding every individual payment
    • among all truthful payment scheme based on this structure
computing payment
Computing Payment
  • Threshold defines a payment
  • Question left: how to find the threshold efficiently?
  • Illustrate for structure
    • Least Cost Path Tree (LCPT) Based
lcpt based method

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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
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

what is cooperative game
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
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
multicast cost sharing fixed tree

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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
shapley value

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3/2+3/2=3

Shapley Value
  • Equally share for downstream receivers
multicast cost sharing valuation
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
with valuation

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With valuation

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cost sharing no fixed tree
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 tree1
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
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
sharing payment lcpt payment

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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
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
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
selfish via game theory
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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