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