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

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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

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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

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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

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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

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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

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Non-cooperative Games
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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

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Example Prisoners Dilemma
1 2 cooperate defect
cooperate (R,R) (S,T)
defect (T,S) (P,P)
Both players view TgtR gt PgtS e.g., T10, R8,
P4, S1
Nash equilibrium (defect,defect)? (p,p)
Global optimum (R,R) if 2RgtTS
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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 prisoners dilemma is
  R/P
• Price of anarchy of a protocol could be large (as
  routing)

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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

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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

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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

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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

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Unicast

• Node vk costs ck to 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

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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

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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!

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VCG Mechanism

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

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Unicast Payment Calculation
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Total payment is 8910 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

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Fast Payment Calculation

• Payment calculation for one node
• Dijstras algorithm
• Time complexity O(n log nm)
• Payment calculation for all nodes on the LCP
• Using Dijstras algorithm for every node
• Time complexity O(n2 log nnm)

• We can calculate it faster!
• Our Result Payment calculation for all nodes on
  the LCP could be done in O(n log nm) 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

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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)

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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

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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

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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
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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

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Payment Optimality

• The payment scheme is optimal regarding every
  individual payment
• among all truthful payment scheme based on this
  structure

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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 graph K(G)
  
• Nodes are receivers, plus source node
• Edges are LCP between two end-points
  
• Find the MST on K(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
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Cooperative Games
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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

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Example Cost Sharing

• Given a set of players N
• The cost of C(S) for every is known
• The cost is cohesive C(ST)lt 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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Shapley Value

• Equally share for downstream receivers

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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

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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!

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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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Sharing payment LCPT payment

• Mechanism using LCPT as output

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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

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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

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Questions and Comments
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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)