Algorithms and Economics of Networks Abraham Flaxman and Vahab Mirrokni, Microsoft Research
Feb 06, 2016
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/education/courses/cse599m/07sp/
Course Work
Scribe One Topic One Problem Set due Mid-May One Project
Contact: {Abie,Mirrokni}@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
Big monkey w
w w
c
cc
0,0
Little monkey
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 we’ll 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, …, si
k
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
Big Monkey
Little Monkey
c
c w
w
5,3 4,4
0,09,1
Nash’s 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: Prisoner’s Dilemma Defect-Defect is the only Nash
equilibrium. It is very bad socially.
cooperate defect
defect 10,0
0,10
1,1
5,5
Row
Column
cooperate
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 Prisoner’s 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 ) = Σi 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
32
2
4
m1 m2
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 ) = Σi 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)
32
2
4
m1 m2
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 => in M, there is a job that wants to switch from j to l.
C(M) ≥ 2 * C(M’) impliesRM( j ) ≥ 2 * RM’( k )
Price of Anarchy for Load Balancing POA for Mixed Nash Equilibria P||C max : for fj(L) = L, POA is 2-2/m+1. Q||C max : for f j (L)=L / s j, POA is
O(logm/loglogm). R||C 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 didn’t talk about Other Equilibrium Concepts: Subgame
Perfect Equilibria, Correlated Equilibria, Cooperative Equilibria
Price of Stability
Next Lecture. Congestion Games
Rosenthal’s Theorem: Congestion games are potential Games:
Market Sharing Games Submodular Games
Vetta’s Theorem: Price of anarchy is ½ for these games.
Selfish Routing Games