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

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Topics Algorithms for Complex Networks Economics and Game Theory

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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.

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Logistics Course Web Page: Course Work Scribe One Topic One Problem Set due Mid-May One Project Contact:

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Why do we study game theory?

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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.

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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.

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

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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.

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Basics of Game Theory

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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.

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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.

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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?

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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.

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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)

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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.

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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…

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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

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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

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Nash’s Theorem Nash defined the concept of mixed Nash equilibria in games, and proved that: Any Strategic Game possess a mixed Nash equilibrium.

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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.

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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

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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.

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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

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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

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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

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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 )

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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.

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

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We didn’t talk about Other Equilibrium Concepts: Subgame Perfect Equilibria, Correlated Equilibria, Cooperative Equilibria Price of Stability

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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

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