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- 1. Introduction Existing Work Contributions Research Proposal Conclusion Game Theoretic Analysis of Network Problems Enoch Lau (Supervisor: Dr Tasos Viglas) Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 2. Introduction Existing Work Contributions Research Proposal Conclusion Project Overview What is this project about? How much worse does a network perform when we allow users to route their traﬃc in a selﬁsh manner? Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 3. Introduction Existing Work Contributions Research Proposal Conclusion Project Overview Context Research themes: Economic notions of game theory Theoretical computer science Putting the two together: The Internet is fertile ground for such analysis Use mathematical tools to reason about the behaviour of systems of users Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 4. Introduction Existing Work Contributions Research Proposal Conclusion Motivating Examples Pigou’s example Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 5. Introduction Existing Work Contributions Research Proposal Conclusion Motivating Examples Pigou’s example Optimal routing: half the traﬃc on each route Selﬁsh routing: everyone goes on the bottom link Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 6. Introduction Existing Work Contributions Research Proposal Conclusion Motivating Examples Pigou’s example Selﬁsh behaviour need not produce a socially optimal outcome. Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 7. Introduction Existing Work Contributions Research Proposal Conclusion The Model Network Model Directed graph with source-destination pairs called commodities Each commodity routes a certain amount of traﬃc, which can be carried over multiple paths Edges have a cost function, which may depend on the amount of ﬂow on the edge (congestion game) Each user controls a negligible amount of ﬂow Economics interpretation in addition to computer science Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 8. Introduction Existing Work Contributions Research Proposal Conclusion The Model Nash Equilibrium Formalises what we mean by selﬁsh behaviour in a game No user can gain by changing strategies unilaterally Stable outcome Pure Nash equilibrium: choose one strategy Mixed Nash equilibrium: choose from a set of strategies with a probability distribution Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 9. Introduction Existing Work Contributions Research Proposal Conclusion The Model How to measure the eﬀects of selﬁsh users? Price of stability: Ratio of cost of best Nash equilibrium to cost of optimal routing Price of anarchy: Ratio of cost of worst Nash equilibrium to cost of optimal routing Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 10. Introduction Existing Work Contributions Research Proposal Conclusion Price of Stability and Anarchy Results Classical Results Nash: every ﬁnite game has a mixed equilibrium (1950) Rosenthal: every congestion game has pure equilibria (1973) Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 11. Introduction Existing Work Contributions Research Proposal Conclusion Price of Stability and Anarchy Results Network Games Roughgarden and Tardos: initiated the price of anarchy in non-atomic network games in 2002 Nash equilibrium at most 33% worse than optimal routing with linear edge cost functions Nash ﬂow is no worse than an optimal ﬂow forced to route twice as much traﬃc Network topology is irrelevant Simplest cases show worst behaviour Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 12. Introduction Existing Work Contributions Research Proposal Conclusion Price of Stability and Anarchy Results Extensions to the Model Changed assumptions about the network: Non-atomic vs. atomic congestion games Splittable vs. unsplittable ﬂow -approximate Nash equilibria Relaxed assumptions on edge cost functions Changed assumptions about the users: Malicious users Oblivious users Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 13. Introduction Existing Work Contributions Research Proposal Conclusion Equilibria for Multicast Routing What is Multicast? Data is sent to multiple recipients, but is sent down each link only once Having multiple users on an edge is now good Congestion games turned on its head Use similar ideas though Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 14. Introduction Existing Work Contributions Research Proposal Conclusion Equilibria for Multicast Routing Price of Anarchy of Multicast Two traditional economics-based edge-cost sharing mechanisms: Shapley value Marginal cost function Economic incentives can be used to encourage optimal behaviour, e.g. taxes Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 15. Introduction Existing Work Contributions Research Proposal Conclusion This Project’s Contributions Reciprocal/Inverse Congestion Games To date, no one has examined decreasing edge-cost functions except in the restricted case of certain economics-based mechanisms Natural interpretations of decreasing edge-cost functions Produce equivalent results to literature, e.g.: Price of stability and anarchy Pure and mixed equilibria Splittable and unsplittable ﬂow Edge capacities Algorithms to compute equilibria Mixed increasing and decreasing functions Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 16. Introduction Existing Work Contributions Research Proposal Conclusion This Project’s Contributions Applications to Multicast Routing Reapply the generalisation to multicast: Generalisation of former charging models could result in a fairer charging scheme Computational and network complexity for the generalised model Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 17. Introduction Existing Work Contributions Research Proposal Conclusion Research Overview Stages of the Project Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 18. Introduction Existing Work Contributions Research Proposal Conclusion Stage 1: Exploration and Experimentation Exploration of Diﬀerent Networks Experimentation is not a key component nor written about in theoretical computer science, but useful to get ideas Find small networks that capture the essence of the problem Systematic exploration of classes of networks: Sparse/dense networks Statistical distributions over traﬃc rates Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 19. Introduction Existing Work Contributions Research Proposal Conclusion Stage 1: Exploration and Experimentation Software AMPL/CPLEX: Optimiser for mathematical programming models expressed in algebraic form GAMUT/Gambit: Generates randomised games and ﬁnds Nash equilibria in restricted cases Mathematica: Visualisation of strategy spaces Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 20. Introduction Existing Work Contributions Research Proposal Conclusion Stage 1: Exploration and Experimentation Stage 1 Risks Cannot ﬁnd generalisations: focus on restricted classes of networks, or make more assumptions about the edge-cost functions Software does not directly solve problem due to violated assumptions: pre or post processing required In general, risks will be identiﬁed by comparison with timeline, and fallbacks initiated if necessary Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 21. Introduction Existing Work Contributions Research Proposal Conclusion Stage 2: Pure Theoretical Results Proof Techniques Linear and convex optimisation Reformulation of programs to reduce exponential size Use of marginal cost function to relate optimal and Nash ﬂows Augmentation of optimal ﬂows to attain Nash ﬂows Lower bounds proved by simple examples Potential functions Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 22. Introduction Existing Work Contributions Research Proposal Conclusion Stage 2: Pure Theoretical Results Stage 2 Risks No guarantee of convexity: many simple mathematical tools such as convex optimisation with KKT conditions ruled out; novel approach in one paper to map to a diﬀerent type of user equilibria No generalisation possible: ﬁnd price of anarchy bounds in restricted network cases Risks minimised by adopting proofs in literature as a template Fallback: report numerical analyses Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 23. Introduction Existing Work Contributions Research Proposal Conclusion Stage 3: Application to Multicast Stage 3 Risks Not core to the project, but good to demonstrate application Main risk is that the generalisation does not match reality Evaluation: analysis of computational and network complexity; re-useable approach outlined in one paper Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 24. Introduction Existing Work Contributions Research Proposal Conclusion Conclusion Summary Selﬁsh users in a network can cause suboptimal global outcomes We turn this on its head and examine what happens when we treat congestion as a good thing Generalise previous work into inverse congestion functions Application in fairer multicast pricing Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 25. Introduction Existing Work Contributions Research Proposal Conclusion Conclusion Acknowledgements I thank Dr Viglas for his supervision thus far. Images: Roughgarden’s book: http://mitpress.mit.edu/catalog/ item/default.asp?tid=10339&ttype=2 Multicast: http://en.wikipedia.org/wiki/Image:Multicast.svg John Nash: http://en.wikipedia.org/wiki/Image: John_f_nash_20061102_3.jpg Robert Rosenthal: http://www.ams.org/featurecolumn/ archive/rationality.html Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems
- 26. Introduction Existing Work Contributions Research Proposal Conclusion Conclusion ∃ p ∈ audience s.t. p has some q ∈ { set of all questions }? Enoch Lau (Supervisor: Dr Tasos Viglas) Game Theoretic Analysis of Network Problems

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