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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

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

- Private cost ci
- Strategies: all possible costs
- Mechanism
- Output O(c): a path
- Payment p(c)
- Node i
- Valuation:
- Utility:

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 Calculation8

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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

1

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

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

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

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

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

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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