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

Preliminaries and Related WorksNon-cooperative games

Price of AnarchyStrategyproof mechanisms for routing

UnicastMulticast

Cooperative gamesCost and payment sharing

Conclusion and future work

Outline

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?

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

1 2

cooperate defect

cooperate (R,R) (S,T)

defect (T,S) (P,P)Both players view: T>R > P>Se.g., T=10, R=8, P=4, S=1Nash 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

Algorithm Mechanism Design

N players Private type ti

Strategy from Ai

Mechanism M=(O,P) Output O(a) Payment p(a)

Player i Valuation: Utility:

),( ii tov

),( iiii tovpu iii cov

N wireless nodes Private cost ci

Strategies: all possible costs

Mechanism Output O(c): a path Payment p(c)

Node i Valuation: Utility: iiii copu

Unicast game

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

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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

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

),,()\,,( GtsLCPvGtsLCPdp kkk

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

Overpayment ratio: %15018

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Unicast Payment Calculation

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

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.

Fast Payment Calculation

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

),( ENG c

)()( xcTC Tx

Multicasting

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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)

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

selectedkp

Not 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

kckc

kckc

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

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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LCPT Based Method

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

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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i i

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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 0q

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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

)()(1

NCxNCn

i i

)(SOPTxSi i

nln

1

nln

1

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!

)()(1

NCxNCn

i i

)(SOPTxSi i

n

1

n

1

n

1

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

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

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

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