CSE 522 – Algorithmic and Economic Aspects of the Internet

1. CSE 522 – Algorithmic and Economic Aspects of the Internet Instructors: Nicole Immorlica Mohammad Mahdian

2. Previously in this class • Properties of social networks • Generative models for power law distribution and power law graphs • Generative models for small-world networks

3. This Lecture Final remarks on small-world networks Network formation games, and a short introduction to game theory

4. Geographic Routing • Experiments suggest that the first criterion that people use for forwarding a message is geographic proximity. • Kleinberg: In a 2-d grid with long-range contact probability proportional to dist –2, “geographic routing” works. • However, experiments show that this probability is closer to dist –1.

5. Geographic Routing, cont’d • Liben-Nowell et al., PNAS 2005: • Justification: in Kleinberg’s model, population is distributed uniformly on a 2-d grid, but in the real world the distribution is not uniform. • Model: probability of a long-range contact from u to v proportional to the inverse of the # of people that are closer to u than v. • Result: In this model, geographic routing works. • Experiments on ~500,000 blogs on LiveJournal confirms the assumption of the model.

6. Getting Closer or Drifting Apart? • Rosenblat and Mobius, QJE 2004: • Technology has made it less costly to interact with people across the globe (‘global village’). • As a result, people become more selective in whom to interact/collaborate with. • Could this fragment the social network into clusters of like-minded people? • Prominent example: scientific community

7. Getting Closer or Drifting Apart? • Model: • Agents of types A and B are arranged uniformly around a circle. • Each person collaborates with a fixed # of other people, and receives a payoff from each collaboration. • The payoff for collaborating with someone of the same type is higher. • Collaborating with someone who is not close has a cost C. • Results: • As C decreases, individual separation (diameter) decreases, but group separation increases. • Experiments on co-authorship among economists (69-99)

8. Network Formation Games • Models that use formal game theoretic reasoning to study network formation • Individuals in a network face economic incentives to form or break links with other individuals • Individuals make self-motivated decisions about which links to form • Applications: professional network, Internet

9. Incentives in Networks • Each individual is a source of benefits (information, resources) • Others can share the benefits of an individual via formation of links • Link formation is costly (time, effort, money) Given these incentives, which links will form?

10. Game Theory Framework • Set of players (agents) • Each player selects a strategy from the set of allowed strategies. • A payoff function specifies how much each player receives given the strategy profile. • An equilibrium is a strategy profile in which no player can benefit by unilaterally changing his strategy.

11. Network Formation Games • Players{1,…,n} are nodes in the network • Each player i must simultaneously choose some subset of {1,…,n} as his strategysi • A strategy profile defines a (directed) graphG • Nodes are players • Edge (i,j) is in G if j 2si

12. Example: Graph • Players = {1, 2, 3, 4} s1 = {4} s2 = {3,4} 1 2 4 3 s2 = {3} s3 = {4}

13. Game Theory Framework • Let Ni = |si| be number of links i forms • Let Cibe “connectedness” of i (definition varies depending on model) • Given a strategy profile (i.e., graph G), the payoff for a player i is a function i(Ni,Ci) decreasing in Niand increasing in Ci • Players seek to maximize their payoff

14. Example: Payoffs 1 = 2 – 1 = 1 2 = 2 – 2 = 0 • E.g., iis number of nodes that i can reach via a directed path in G minus the number of links i forms 1 2 4 3 3 = 1 – 1 = 0 3 = 1 – 1 = 0

15. Equilibrium Networks When do we expect a graph to be stable? • A graph G is a Nash equilibrium if no player has an incentive to unilaterally sever or create links, i.e. for any other strategy s’iof i, his payoff ’iin the resulting graph G’ is at most his payoff iin G

16. Example: Equilibrium Networks 1 = 2 – 1 = 1 ’1 = 3 – 1 = 2 2 = 2 – 2 = 0 • Node 1 has an incentive to sever connection to 4 and instead form a connection to 2 for a resulting payoff of ’1 = 3 – 1 = 2 1 2 4 3 3 = 1 – 1 = 0 3 = 1 – 1 = 0

17. Strict Equilibria What if there is another strategy for a player which does not change his payoff? • A graph G is a strict Nash equilibrium if each player’s strategy is his unique best-response, i.e. for any other strategy s’iof i, his payoff ’iin the resulting graph G’ is strictly less than his payoff iin G

18. Example: Strict Equilibria 2 = 3 – 1 = 2 1 = 3 – 1 = 2 • Any unilateral deviation by a node strictly decreases his payoff 1 2 4 3 3 = 3 – 1 = 2 3 = 3 – 1 = 2

19. Models: Bala and Goyal • Two models (Bala and Goyal, Econometrica 2000) • One-way flow: A link can be used only by the person who formed it to send information • Two-way flow: A link between two people can be used by either person • Model is frictionless if value of information does not decay with distance: Ciis number of nodes i can reach in G by a path of any length

20. Equilibria in Bala and Goyal • For any payoff function • In both models, every Nash equilibrium is either connected or empty • In the one-way flow model, the only strict Nash equilibria are the directed cycle and/or the empty network • In the two-way flow model, the only strict Nash equilibria are the center-sponsored star (one node connects to all others) and/or the empty network

21. Experimentation: Falk and Kosfeld • Implemented game with 4 players • Players were offered 10 points (worth 65 cents each) for each player they had a direct or indirect connection to (including themselves) • Players were charged C points for each link they formed • There were five treatments: C = 5, 15, and 25 in one-way model and C = 5 and 15 in two-way model

22. Predictions vs Results • Explanations: • Symmetry of strategies/coordination issue • Inequity aversion (people prefer equal payoffs) • Concern for efficiency (empty graph gives no payoffs)

23. Dynamics in Bala and Goyal • Does not imply that equilibria are unique! For example, there are n possible stars. Can players find an equilibria? • Consider following best-response dynamic • Start from an arbitrary initial graph • In each period, each player independently decides to “move” with probability p • If a player decides to move, he picks a new strategy randomly from his set of best responses to graph in previous period

24. Dynamics in Bala and Goyal • Theorem: In either model, the dynamic process converges to a strict Nash equilibrium network with probability one. • Simulations show that rate of convergence is quite rapid.

25. Accounting for Distances • Bala and Goyal • Value of information decays by a factor of for each link traversed (model with “friction”) • Results similar to frictionless models still hold • Fabrikant et al. (PODC 2003) • Value of connection to j for a node i is -d(i,j) (and payoff function is linear) • Nash equilibria become slightly more complex (e.g., trees are Nash equilibria in some cases)

26. Model: Fabrikant et al. • The payoff incurred by player i is i = -  Ni – jd(i,j) where Niis the number of links formed by i and d(i,j) is the distance between i and j in the underlying undirected network (two-way flow model).

27. Equilibria: Fabrikant et al. • For  < 1, complete graph is only Nash equilibrium • For  > n2, all Nash equilibria are trees • Conjecture: For  > some constant, all strict Nash equilibria are trees. • Upcoming paper in SODA 2006 disproves this.

28. Efficiency of the Equilibria • The social welfare or efficiency of a strategy profile in a game is defined as the sum of payoffs of all players • The price of anarchy of a game is the ratio of least-efficient Nash equilibrium to the most-efficient strategy profile (which need not be an equilibrium) • Theorem [Fabrikant et al.]: For any tree Nash equilibrium T, the welfare of T to the optimum is at most 5.

29. Other Network Formation Games What if agents cooperate to form links?

30. Cooperative Game Theory • Players cooperate to achieve a common goal (e.g., building a network). • Achieving goal has a value for each agent. • In a transferable utility game, agents must additionally decide how to share this value among each other. • As in non-cooperative game theory, analyze stable situations, but now must consider coalitions as well as individuals.

31. Cooperative Network Formation • Jackson and Wolinsky • Studied network formation as a cooperative game with transferable utilities, in particular individuals can share cost of links. • Showed there are natural situations in which no efficient network is pairwise stable for any utility-transfer rule.