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: Course Work  Scribe One Topic  One Problem Set due Mid-May  One Project Contact: 

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 c c 0,0 Little monkey 9,16-2,47-2,3 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): s i 1, …, s i k  strategy profile: a vector of strategies (one for each player)  for each strategy profile s, a payoff P i s 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 cw 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. cooperatedefect 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 w i m strategies/machines Outcome M: assignment jobs → machines J( j ): jobs on machine j L( j ) = Σ i in J( j ) w i : 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 m 1 m 2

Load Balancing Games (parts of slides from E. Elkind, warwick) n players/jobs, each with weight w i m strategies/machines Outcome M: assignment jobs → machines J( j ): jobs on machine j L( j ) = Σ i in J( j ) w i : 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 max j R(j) m 1 m 2

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 max j R(j) Theorem: if f j (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 )=R M ( j )  k: worst machine in M’, i.e., C( M’ )=R M’ ( k )  there is an l s.t. R M ( l ) ≤ R M’ ( k ) (averaging argument)  w = max w i ; R M’ ( k ) ≥ w  R M ( j ) - R M ( l ) ≥ 2R M’ ( k ) - R M’ ( k ) ≥ w => in M, there is a job that wants to switch from j to l. C(M) ≥ 2 * C(M’) implies R M ( j ) ≥ 2 * R M’ ( k )

Price of Anarchy for Load Balancing POA for Mixed Nash Equilibria P||C max : for f j (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 f j (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