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Algorithms and Economics of Networks

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Topics Algorithms for Complex Networks Economics and Game Theory Algorithms for Large Networks TraceRoute Sampling Where do networks come ... Load Balancing Games. – PowerPoint PPT presentation

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Title: Algorithms and Economics of Networks


1
Algorithms and Economics of Networks
  • Abraham Flaxman and Vahab Mirrokni, Microsoft
    Research

2
Topics
  • Algorithms for Complex Networks
  • Economics and Game Theory

3
Algorithms for Large Networks
  • TraceRoute Sampling
  • Where do networks come from?
  • Network Formation
  • Link Analysis and Ranking
  • What Can Link Structure Tell Us About Content?
  • Hub/Authority and Page-Rank Algorihtms
  • Clustering 
  • Inferring Communities from Link Structure
  • Local Partitioning Based on Random Walks
  • Spectral Clustering
  • Balanced Partitioning.
  • Diffusion and Contagion in Networks Spread of
    Influence in Social Networks.
  • Rank Aggregation   
  • Recent Algorithmic Achievements.

4
Logistics
  • Course Web Page http//www.cs.washington.edu/educ
    ation/courses/cse599m/07sp/
  • Course Work
  • Scribe One Topic
  • One Problem Set due Mid-May
  • One Project
  • Contact
  • Abie,Mirrokni_at_Microsoft.com

5
Why do we study game theory?
6
Selfish Agents
  • Many networking systems consist of
    self-interested or selfish agents.
  • Selfish agents optimize their own objective
    function.
  • Goal of Mechanism Design encourage selfish
    agents to act socially.
  • Design rewarding rules such that when agents
    optimize their own objective, a social objective
    is met.

7
Self-interested Agents
  • How do we study these systems?
  • Model the networking system as a game, and
  • Analyze equilibrium points.
  • Compare the social value of equilbirim points to
    global optimum.

8
Algorithmic Game Theory
  • Important Factors
  • Existence of equilibria as as subject of study.
  • Performance of the output (Approximation Factor).
  • Convergence (Running time) ? Computer Science

9
Economics of Networks
  • Lack of coordination in networks    
  • Equilibrium Concepts Strategic Games and Nash
    equilibria
  • Price of Anarchy.
  • Load Balancing Games.
  • Selfish Routing Games and Congestion Games.
  • Distributed Caching and Market Games.
  • Efficiency Loss in Bandwidth Allocation Games.
  • Coordination Mechanisms
  • Local Algorithmic Choices Influence the Price of
    Anarchy.
  • Market Equilibria and Power Assignment in
    Wireless Networks.
  • Algorithms for Market Equilibria.
  • Power Assignment for Distributed Load Balancing
    in Wireless Networks. 
  • Convergence and Sink Equilibria
  • Best-Response dynamics in Potential games.
  • Sink Equilibria Outcome of the Best-response
    Dynamics.
  • Best response Dynamics in Stable Matchings.

10
Basics of Game Theory
11
Game Theory
  • Was first developed to explain the optimal
    strategy in two-person interactions
  • Initiated for Zero-Sum Games, and two-person
    games.
  • We study games with many players in a network.

12
Example Big Monkey and Little Monkey
  • Example by Chris Brook, USFCA
  • Monkeys usually eat ground-level fruit
  • Occasionally climb a tree to get a coconut (1 per
    tree)
  • A Coconut yields 10 Calories
  • Big Monkey spends 2 Calories climbing the tree.
  • Little Monkey spends 0 Calories climbing the
    tree.

13
Example Big Monkey and Little Monkey
  • If BM climbs the tree
  • BM gets 6 C, LM gets 4 C
  • LM eats some before BM gets down
  • If LM climbs the tree
  • BM gets 9 C, LM gets 1 C
  • BM eats almost all before LM gets down
  • If both climb the tree
  • BM gets 7 C, LM gets 3 C
  • BM hogs coconut
  • How should the monkeys each act so as to maximize
    their own calorie gain?

14
Example Big Monkey and Little Monkey
  • Assume BM decides first
  • Two choices wait or climb
  • LM has also has two choices after BM moves.
  • These choices are called actions
  • A sequence of actions is called a strategy.

15
Example Big Monkey and Little Monkey
c
w
Big monkey
c
w
c
w
Little monkey
0,0
9,1
6-2,4
7-2,3
  • What should Big Monkey do?
  • If BM waits, LM will climb BM gets 9
  • If BM climbs, LM will wait BM gets 4
  • BM should wait.
  • What about LM?
  • Opposite of BM (even though well never get to
    the right side of the tree)

16
Example Big Monkey and Little Monkey
  • These strategies (w and cw) are called best
    responses.
  • Given what the other guy is doing, this is the
    best thing to do.
  • A solution where everyone is playing a best
    response is called a Nash equilibrium.
  • No one can unilaterally change and improve
    things.
  • This representation of a game is called extensive
    form.

17
Example Big Monkey and Little Monkey
  • What if the monkeys have to decide
    simultaneously?
  • It can often be easier to analyze a game through
    a different representation, called normal form
  • Strategic Games One-Shot Normal-Form Games with
    Complete Information

18
Normal Form Games
  • Normal form game (or Strategic games)
  • finite set of players 1, , n
  • for each player i, a finite set of actions (also
    called pure strategies) si1, , sik
  • strategy profile a vector of strategies (one for
    each player)
  • for each strategy profile s, a payoff Pis to each
    player

19
Example Big Monkey and Little Monkey
  • This Game has two Pure Nash equilibria
  • A Mixed Nash equilibrium Each Monkey Plays each
    action with probability 0.5

Little Monkey
c
w
5,3
4,4
c
Big Monkey
w
0,0
9,1
20
Nashs Theorem
  • Nash defined the concept of mixed Nash equilibria
    in games, and proved that
  • Any Strategic Game possess a mixed Nash
    equilibrium.

21
Best-Response Dynamics
  • State Graph Vertices are strategy profiles. An
    edge with label j correspond to a strict
    improvement move of one player j.
  • ? Pure Nash equilibria are vertices with no
    outgoing edge.
  • Best-Response Graph Vertices are strategy
    profiles. An edge with label j correspond to a
    best-response of one player j.
  • Potential Games There is no cycle of strict
    improvement moves ? There is a potential function
    for the game.
  • BM-LM is a potential game. Matching Penny game is
    not.

22
Example Prisoners Dilemma
  • Defect-Defect is the only Nash equilibrium.
  • It is very bad socially.

Column
cooperate
defect
0,10
5,5
cooperate
Row
defect
1,1
10,0
23
Price of Anarchy
  • The worst ratio between the social value of a
    Nash equilibrium and social value of the global
    optimal solution.
  • An example of social objective the sum of the
    payoffs of players.
  • Example In BM-LM Game, the price of anarchy for
    pure NE is 8/10. POA for mixed NE is 6.5/10.
  • Example In Prisoners Dilemma, the price of
    anarchy is 2/10.

24
Load Balancing Games
  • n players/jobs, each with weight wi
  • m strategies/machines
  • Outcome M assignment jobs ? machines
  • J( j ) jobs on machine j
  • L( j ) Si in J( j ) wi load of j
  • R( j ) f j ( L( j ) ) response time of j
  • f j monotone, 0
  • e.g., f j (L)L / s j
  • (s j is the speed of machine j)
  • NE no job wants to switch, i.e., for any i in J(
    j )
  • f j ( L( j ) ) f k ( L( k ) w j )
    for all k ? j

25
Load Balancing Games(parts of slides from E.
Elkind, warwick)
  • n players/jobs, each with weight wi
  • m strategies/machines
  • Outcome M assignment jobs ? machines
  • J( j ) jobs on machine j
  • L( j ) Si in J( j ) wi load of j
  • R( j ) f j ( L( j ) ) response time of j
  • f j monotone, 0
  • e.g., f j (L)L / s j
  • (s j is the speed of machine j)
  • NE no job wants to switch, i.e., for any i in J(
    j )
  • f j ( L( j ) ) f k ( L( k ) w j )
    for all k ? j
  • Social Objective worst response time maxj R(j)

26
Load Balancing Games
  • Theorem if all response times are nonegative
    increasing functions of the load, pure NE exists.
  • Proof
  • start with any assignment M
  • order machines by their response times
  • allow selfish improvement reorder
  • each assignment is lexicographically better than
    the previous one

jobs migrate from left to right
27
Load Balancing Games POA
  • Social Objective worst response time maxj R(j)
  • Theorem if fj(L) L (response time load),
    Worst Pure Nash/Opt 2.
  • Proof
  • M arbitrary pure Nash, M Opt
  • j worst machine in M, i.e., C( M )RM( j )
  • k worst machine in M, i.e., C( M )RM( k )
  • there is an l s.t. RM( l ) RM( k ) (averaging
    argument)
  • w max wi RM( k ) w
  • RM( j ) - RM( l ) 2RM( k ) - RM( k ) w
    gt
  • in M, there is a job that wants to switch
    from j to l.

C(M) 2 C(M) implies RM( j ) 2
RM( k )
28
Price of Anarchy for Load Balancing
  • POA for Mixed Nash Equilibria
  • PC max for fj(L) L, POA is 2-2/m1.
  • QC max for f j (L)L / s j, POA is
    O(logm/loglogm).
  • RC max for fj(L) L and each job can be
    assigned to a subset of machines, POA is
    O(logm/loglogm).
  • Will give some proofs in the lecture on
    coordination mechanisms.

29
We Know
  • Normal Form Games
  • Pure and Mixed Nash Equilibria
  • Best-Response Dynamics, State Graph
  • Potential Games
  • Price of Anarchy
  • Load Balancing Games

30
We didnt talk about
  • Other Equilibrium Concepts Subgame Perfect
    Equilibria, Correlated Equilibria, Cooperative
    Equilibria
  • Price of Stability

31
Next Lecture.
  • Congestion Games
  • Rosenthals Theorem Congestion games are
    potential Games
  • Market Sharing Games
  • Submodular Games
  • Vettas Theorem Price of anarchy is ½ for these
    games.
  • Selfish Routing Games
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