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Algorithms and Economics of Networks

- Abraham Flaxman and Vahab Mirrokni, Microsoft

Research

Topics

- Algorithms for Complex Networks
- Economics and Game Theory

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.

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

Why do we study game theory?

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.

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.

Algorithmic Game Theory

- Important Factors
- Existence of equilibria as as subject of study.
- Performance of the output (Approximation Factor).
- Convergence (Running time) ? Computer Science

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.

Basics of Game Theory

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.

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.

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?

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.

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)

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.

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

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

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

Nashs Theorem

- Nash defined the concept of mixed Nash equilibria

in games, and proved that - Any Strategic Game possess a mixed Nash

equilibrium.

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.

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

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.

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

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)

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

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 )

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.

We Know

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

We didnt talk about

- Other Equilibrium Concepts Subgame Perfect

Equilibria, Correlated Equilibria, Cooperative

Equilibria - Price of Stability

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