1. Price of Anarchy for the N-player Competitive Cascade Game with Submodular Activation Functions Xinran He, David Kempe {xinranhe, dkempe}@usc.edudkempe}@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 a social network. Competitive Diffusion in the social network

4. Competitive cascade 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

7. General adoption model

8. General adoption model: Local Dynamic

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

10. Useful properties = Prob{ } ? ?

11. Price of anarchy

12. Main results Improvement over [Goyal/Kearns 2012]: Improve PoA upper bound from 4 to 2. Generalize result from 2 player game to N player game.

13. Proof roadmap Set Game Valid utility game PoA bounds By reduction to general threshold model By global competitiveness By definition of social utility function [Vetta 2002] [Roughgarden 2009]

14. Proof roadmap Set Game Valid utility game PoA bounds By reduction to general threshold model By global competitiveness By definition of social utility function [Vetta 2002] [Roughgarden 2009] By definition.

15. Proof roadmap Set Game Valid utility game PoA bounds By reduction to general threshold model By global competitiveness By definition of social utility function [Vetta 2002] [Roughgarden 2002]

17. Update sequence: Active Inactive

18. Update sequence: Active Inactive

19. Proof roadmap Set Game Valid utility game PoA bounds By reduction to general threshold model By global competitiveness By definition of social utility function [Vetta 2002] [Roughgarden 2009]

21. Proof: wrap up The competitive cascade game is a valid utility game The pure PoA is bounded by 2 [Vetta 2002] The coarse PoA is bounded by 2 [Roughgarden 2009]

22. Tightness of upper bound

23. Conclusion

24. Future work