Price of Anarchy for the N-player Competitive Cascade Game with S ubmodular Activation Functions

1. Price of Anarchy for the N-player Competitive Cascade Game with SubmodularActivation Functions Xinran He, David Kempe {xinranhe, dkempe}@usc.edu 12/14/2013

2. Diffusion In Social Network • The adoption of new products can propagate in the social network Diffusion in the social network

3. Competitive Diffusion In Social Network • Different products compete for acceptance in asocial network. • Competitive Diffusion in the social network

4. Competitive cascade game • Given a social network • The players are N companies, with their products . • The individuals can be in state and . • The players simultaneously allocate resources to individuals in the social network in order to seed them as initial adopters of their products. • The adoption of products propagates according to diffusion model. • The goal for each player is to maximize the coverage of his own product. • In this paper, we study the Price of Anarchy of this game.

5. Main contribution The upper bound on the coarse Price of Anarchy is 2 for the N player competitive cascade game under the Goyal/Kearns diffusion model. • Improvement over [Goyal/Kearns 2012]: • Improve PoA upper bound from 4 to 2. • Generalize result from 2 player game to N player game. • Simple and clear proof by resorting to valid utility game and general threshold model.

6. Competitive cascade game • Given a social network . • N players, each player is a company with limit budget . • Strategy vector for players: • is the set of nodes selected by company . • . • Payoff function: • Expected number of people who adopt product . • Social utility function : • Expected number of people who adopt a product.

7. General adoption model • Seeding stage: • Each company selects a set of individuals . • The initial state of node is inactive if no company selects it. • Otherwise, the node becomes in state uniformly at random. • Diffusion stage: • Given a fixed update sequence . • Nodes change states with the order in according to local dynamics.

8. General adoption model: Local Dynamic • Let be current sets of nodes in state . • Adoption function: • = Prob{ adopts product } • Total activation probability: • A still inactive node changes into states with probability , and remains inactive with probability .

9. General adoption model: Example D G D C B D F F C C END A E

10. Useful properties Additivity of total activation probability , activation function is monotone. Prob{ } Prob{ } Prob{ } Prob{ } ? ? Submodularity of activation function: = Competitivenessof adoption function:

11. Main results Theorem: Assume the following conditions hold: • For every node , the total activation probability is additive. • For every node , the activation function is submodular. • For every player and node , the adoption function is competitive. Then, the upper bound on the coarse PoA is 2 in the competitive cascade game. • Improvement over [Goyal/Kearns 2012]: • Improve PoA upper bound from 4 to 2. • Generalize result from 2 player game to N player game.

12. Proof roadmap [Vetta2002] PoA bounds Valid utility game Set Game [Roughgarden2009]

13. Proof roadmap [Vetta2002] PoA bounds Valid utility game Set Game [Roughgarden2009] By definition.

14. Proof roadmap [Vetta2002] PoA bounds Valid utility game Set Game [Roughgarden 2002]

15. Submodular : General Threshold model • General Threshold (GT) Model [KKT 03] • Each node has a threshold uniform in [0,1] • Each node has an activation function, is the set of activated nodes. • A node becomes active if and only if . • is expected number of activated nodes at the end of the process. Theorem [Mossel/Roch 2007]: Under the general threshold model with monotone and submodular, σ(S) is monotone and submodular.

16. Submodular : reduction to GT model Active Inactive Update sequence:

17. Proof roadmap [Vetta2002] PoA bounds Valid utility game Set Game [Roughgarden2009]

18. Proof of • Global competitiveness: • Similar to Lemma 1 in [Goyal/Kearns 2012] • Couple two process with and with . • By induction,

19. Proof: wrap up Lemma: and is competitive. Lemma: social utility function is submodular, if is additive and is submodular. The competitive cascade game is a valid utility game The pure PoAis bounded by 2 [Vetta 2002] The coarse PoAis bounded by 2 [Roughgarden 2009]

20. Conclusion • Improvement over [Goyal/Kearns 2012]: • Improve PoA upper bound from 4 to 2. • Generalize from 2 players to N players. • Generalize from pure PoA to coarsePoA. • With a much simpler and clear proof. • Further extensions: • Strategy as multiset: • Budget limit on nodes: • Different node weight ,

21. Future work • Open question • What is the PoA upper bound for competitive cascade game without submodularity of activation function? • Upper bound 4 with additive total activation probability and competitive adoption function for 2 player games. [Goyal/Kearns 2012] • Lower bound 2 by simple example. • Results on cascade without submodularity • Influence maximization: • Single product: submodularity -> . [KKT 2003] • Competitive cascade game

22. Questions?