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/education/courses/cse599m/07sp/ • Course Work • Scribe One Topic • One Problem Set due Mid-May • One Project • Contact: • {Abie,Mirrokni}@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 Little monkey w 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 we’ll 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 payoffPis 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 9,1 0,0

20. Nash’s 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: Prisoner’s Dilemma • Defect-Defect is the only Nash equilibrium. • It is very bad socially. Column cooperate defect cooperate 5,5 0,10 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 Prisoner’s Dilemma, the price of anarchy is 2/10.

24. 2 2 4 3 m1 m2 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 jis 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. 2 2 4 3 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 jis 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 => 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 • 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.

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

31. 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