1. Distributed Computing with Adaptive Heuristics Michael Schapira Princeton Innovations in Computer Science 09 January 2011 Partially supported by NSF Aaron D. Jaggard Rutgers/Colgate Rebecca N. Wright Rutgers

2. This Talk Identify new aspects of the boundary between game theory and distributed computing Look at some initial results Identify various avenues for future work

3. nodes interacting in an asynchronous environment simple, myopic rules of behavior; convergence in synchronized environment Background Areas

4. Motivation Many real-world settings involve both simple, myopic behavior and asynchronicity – Network protocols Routing Congestion control... – Asynchronous circuits

5. Goal: Explore dynamics of adaptive heuristics when asynchrony is allowed Research Agenda novel applications in distributed computing (congestion control, asynchronous circuits,…) new questions in game theory and economics (fictitious play, regret minimization,…)

6. Understanding when Dynamics Converge: A Simple Example Two stable states (pure Nash equilibria) (5,4) (1,1) (0,0) (4,5) Bach Stravinsky BachStravinsky

7. Understanding when Dynamics Converge: A Simple Example If either player is activated alone, the system converge (5,4) (1,1) (0,0) (4,5) Bach Stravinsky BachStravinsky

8. Understanding when Dynamics Converge: A Simple Example Without control over who is activated, the system might not converge (5,4) (1,1) (0,0) (4,5) Bach Stravinsky BachStravinsky

9. Basic Model n nodes (the players) Node i has action space A i Each node i has a reaction function f i : A 1 x A 2 x...x A n →A i that determines i’s next action based on other current actions – No dependence on own action

10. Dynamics Infinite sequence of discrete timesteps t = 1, … Schedule s:{1,…} → 2 [n] determines which set of players is activated at time t. – Fair schedules Start at an initial state; at each time step t, let the nodes in s(t) react using their reaction functions

11. Convergence The players’ action profile a=(a 1,…, a n ) is a stable state if f i (a) = a i for every i. The system is convergent if the dynamics always converge (for all initial states and all fair schedules)

12. Two High-Level Questions What classes of systems are guaranteed (or cannot be guaranteed) to always converge to a stable state? How hard is it to determine whether a system always converges to a stable state?

13. Basic Result Theorem: If the system has multiple stable states, then the system is not convergent. (I.e., there is some initial state and some schedule that diverge.) Actually, can strengthen this: – Allow some history-dependence – Allow randomness in reaction functions

14. Revisiting Our Example (5,4) (1,1) (0,0) (4,5) Bach Stravinsky BachStravinsky

15. A Few Words About the Proof Inspired by approach to FLP result on impossibility of resilient consensus

16. Applications Interdomain routing Congestion control Best-reply dynamics in general games Diffusion of technologies in social networks Asynchronous circuits …

17. Communication Complexity Theorem: Determining whether a system of n nodes, each with two actions, is convergent may require Ω(2 n ) bits. – Even if all reaction functions are deterministic, and do not depend on history or own action Uses a reduction from SET DISJOINTNESS. – Constructed system has a unique stable state

18. Computational Complexity Theorem: Determining whether a system of n nodes, each with deterministic and historyless reaction function, is convergent is PSPACE- complete. So, difficult even if the reaction functions are succinctly represented (so that they could be transmitted quickly) Under complexity assumptions, no short witnesses (in general) of being convergent

19. Scheduling Question: Does randomness help? (5,4) (1,1) (0,0) (4,5) Bach Stravinsky BachStravinsky

20. Scheduling Question: Does randomness help? Answer: No. Divergence may not only be possible, but overwhelmingly likely. Issues of r-fairness

21. Open Questions What are the convergence guarantees and impossibility results – For other heuristics – For other notions of convergence – For other notions of equilibrium We’ve taken first steps in the context of regret-minimizing dynamics

22. Other Open Questions Variations in information – Outdated information – Knowledge only of own utility function (uncoupled dynamics) Lots of others

23. Summary Simple behaviors show up in lots of settings – Important to understand dynamic behavior when asynchrony is allowed Initial results on the impossibility of guaranteeing convergence Lots of open questions – What can we say about the dynamic behavior in other natural asynchronous settings?