Approximation Algorithms and Games on Networks

1
Approximation Algorithms and Games on Networks

• Éva Tardos
• Cornell University

2
Interacting selfish users

• Internet
• Users with a multitude of diverse economic
  interests
  
• browsers
• routers
• servers
• Selfishness
• Parties will deviate from their protocol if it is
  in their interest
  
• Study resulting issues
• Algorithmic game theory
• algorithms game theory

3
Few Algorithmic Issues

• Price of Anarchy
• Measure degradation of performance caused by lack
  of cooperation (selfishness)
  
• Mechanism Design
• How to design games so that selfish behavior
  leads to desired outcome
  
• Coalitional Games
• E.g., how to share cost incurred by a group of
  users,

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Solution Concept?

• What happens when selfish users interact?
• Nash equilibrium (randomized)
• or double best response
• if everyone plays Nash equilibrium no incentive
  to deviate
  
• Theorem Nash 1952 Always exists
• Critique
• can users learn the best behavior? Greenwald,
  Friedman and Shenker
  
• There can be many Nash equilibriums

. . .
5
Other concepts of rationality

• (weakly) dominating strategy
• ( duh?)
• problem too strong, rarely exists
• Example
• Vickery Auction (second prize)
• Revealing the true value is a dominating strategy!

. . .
6
Algorithmic Mechanism Design

• agents have utilities (or values) but these
  utilities are known only to them
  
• game designer prefers certain outcomes depending
  on players utilities
  
• Mechanism DesignDesign game (mechanism) so that
  
  
• players have (weakly) dominating strategies
• selfish behavior leads to outcome desired by
  designer

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Algorithmic Mechanism Design

• Strong results for social welfare maximization
  (VCG)
  
• Auctions
• Buy edges to form a spanning tree
• Buy edges to form an (s,t)-paths

shortest alternate
s
t

• VCG mechanism often breaks the bank
• e.g. for (s,t)-paths
• Archer and Tardos SODA 02

8
Price of Anarchy

• Papadimitriou-Koutsoupias 99
• This talk price of anarchy in routing
  Roughgarden-Tardos FOCS00
  
• Also Spirakis and Mavronikolas 01,
• Roughgarden 01-02,
• Koutsoupias and Spirakis 01,
• Czumaj and Vöcking 02
• Friedman 02

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Traffic in Congested Networks

• Mathematical model
• A directed graph G (V,E)
• sourcesink pairs si,ti for i1,..,k
• rate ri ? 0 of traffic between si and ti for each
  i1,..,k
  
• For each edge e, a latency function le()

r1 1
le(x)x
s1
t1
le(x)1
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Flows and Their Cost

• Traffic and Flows
• A flow vector f specifies a traffic pattern
• fP amount routed on si-ti path P

lP(f) .5 0 1

• The Cost of a Flow
• lP(f) sum of latencies of edges along P
  (w.r.t. flow f)
  
• C(f) cost or total latency of a flow f ?P fP
  lP(f)

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Example
Traffic rate r 1, k 1
x
Cost of flow .5.5 .51 .75
Flow .5
s
t
1
Flow .5
But traffic on lower edge is envious.
An envy free flow
Cost of flow 11 01 1
x
Flow 1
s
t
1
Flow 0

• Agents are selfish want to
• minimize personal latency,
• do not care about welfare of others

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Flows and Game Theory

• Flow represents routes of many noncooperative
  agents
  
• each agent controlling infinitesimally small
  amount
  
• cars in a highway system
• packets in a network
• the cost (total latency) of a flow represents
  social welfare
  
• agents are selfish want to
• minimize personal latency,
• do not care about social welfare

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Flows at Nash Equilibrium

• A flow is at Nash equilibrium (or is a Nash flow)
  if no agent can improve its latency by changing
  its path
  
• Assumption edge latency functions are
  continuous, and non-decreasing
  
• Lemma a flow f is at Nash equilibrium if and
  only if all flow travels along minimum-latency
  paths between its source and destination (w.r.t.
  f)
• Theorem Beckmann et al 56 The Nash equilibrium
  exits and is essentially unique

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Cost of Selfishness

• Cost of flow (total latency) of a flow
  represents social welfare
  
• Our Question To what extent does a Nash flow
  optimize social welfare?
  
• Papadimitriou-Koutsoupias 99

x
1
.5
s
t
1
0
.5
Cost of Nash flow 11 01 1
Cost of optimal (min-cost) flow .5.5
.51 .75
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Braesss Paradox
Traffic rate r 1
Cost of Nash flow 1.5
Cost of Nash flow 2 All the flow has increase
d delay!
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Our Results

• Theorem 1
• In a network with linear latency functions
• i.e., of the form le(x)aexbe
• the cost of a Nash flow is at most 4/3 times that
  of the minimum-latency flow

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General Latency Functions

• Question what about more general edge latency
  functions?
  
• Bad Example (r 1, i large)

xi
1
1-?
s
t
1
0
?
A Nash flow can cost arbitrarily more than the
optimal (min-cost) flow
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Our Results

• Theorem 2
• In any network with continuous, nondecreasing
  latency functions
  
• the cost of a Nash flow with rates ri for
  i1,..,k
  
• is at most the cost of an optimal flow with rates
  2ri for i1,..,k
  

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Characterizing the Min-Cost Flow

• Min-latency flow
• for one s-t pair for simplicity
• minimize C(f) ?e fe le(fe)
• subject to f is an s-t flow
• carrying r units
• By summing over edges rather than paths where fe
  amount of flow on edge e
  
• Convex program if le(fe) convex
• For example, if le(fe) ae fe be then
• C(f) ?e fe (ae fe be) convex quadratic

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Characterizing the Optimal Flow

• Optimality condition moving a tiny flow from one
  path to another cannot decrease the cost
  
• gradient of a path P marginal cost of
  increasing flow along P

flow f is optimal if and only if all flow travels
along minimum-gradient paths
Recall a flow f is at Nash equilibrium if and
only if all flow travels along minimum-latency
paths
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Consequence for Linear Latency Fns

• Observation if le(fe) ae fe be
• (latency functions are linear) ? gradient of P
  w.r.t. f is
  
• ? 2ae fe be
• latency of P w.r.t. f is
• ? ae fe be
• Corollary
• f a Nash flow with rate r ? f/2 is optimal with
  rate r/2

e?P
e?P
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Bounding the Nash

• Goal prove that a Nash flow f costs at most 4/3
  an opt flow in networks with linear latency.

• We prove
• opt at rate r/2 is f/2 ? cost
  ? ¼ C(f)
  
• cost of augmenting from
• rate r/2 to r ? ½ C(f)

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Cost of Augmentation

• Consider the Nash flow f
• let L be the s-t latency in f
• ? Cost of flow f is C(f) rL

L 2

• Recall opt flow at rate r/2 is flow f/2,
  gradient along flow paths in flow is L
  
• ? the marginal cost of increasing flow from
  flow f/2 is L
  
• ? Cost of increasing flow amount by r/2 at
  least (r/2) L ½ C(f)

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Nonlinear Latency

• Theorem 2 In any network with continuous,
  nondecreasing latency functions, the cost of a
  Nash flow with rate r is at most the cost of the
  optimal flow with rate 2r
• Analogous proof sketch??

Troubles
Can be close to zero
What opt at rate r is, and what is its gradient?
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Proof Sketch for Nonlinear Latency

• Augmenting Nash to Opt?
• Idea
• Gradient is at least the latency!!
• Marginal cost to increase Nash?
• But Nash can be improved!
• Idea Separate effects of increased and decreased
  flow

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Other Models?

• An approximate version of Theorem for non-linear
  latency with imprecise evaluation of path
  latency
  
• Analogue for the case of finitely many agents
  (splittable flow)
  
• Extends to other nonatomic congestion games
• Impossibility results for finitely many agents,
  unsplittable flow, i.e.,
  
• if each agent i controls a positive amount of
  flow ri ? 0
  
• flow of a single agent has to be routed on a
  single path

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Algorithmic Game Theory

• The main ingredients
• Lack of central control
• like distributed computing
• Selfish participants
• game theory
• Common in many settings e.g., Internet
• Exciting new area with many open problems
• Cost of anarchy in other network games
• Design networks with low cost of anarchy