1. An Improved Liar Game Strategy From a Deterministic Random Walk Robert Ellis February 22 nd, 2010 Peled Workshop, UIC Joint work with Joshua Cooper, University of South Carolina

2. Genealogy  Uri Peled -> Peter Hammer -> Marian Kwapisz -> ? -> Wacław Pawelski -> Tadeusz Ważewski -> Stanislaw Zaremba -> Gaston Darboux -> Michel Chasles <- H.A. Newton <- E.H. Moore <- George Birkhoff <- Hassler Whitney <- Herbert Robbins <- Herbert Wilf <- Fan Chung <- Robert Ellis 6 th cousins once removed? Peled number <= 4: Peled -> Harary -> Erdős -> Chung -> Ellis 2

3. Outline  Diffusion processes on Z –Simple random walk (linear machine) –Liar machine –Pointwise and interval discrepancy  Pathological liar game –Definition –Reduction to liar machine –Sphere bound and comparisons  Improved pathological liar game bound  Concluding remarks 3

4. 9-9-8-7-6-5-4-3-2012345678 4 11 Linear Machine on Z

5. 9-9-8-7-6-5-4-3-2012345678 Linear Machine on Z 5.5

6. 9-9-8-7-6-5-4-3-2012345678 Linear Machine on Z 2.755.52.75 Time-evolution: 11 £ binomial distribution of {-1,+1} coin flips

7. Liar Machine on Z 9-9-8-7-6-5-4-3-2012345678 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 11 chips t=0 Approximates linear machine Preserves indivisibility of chips

8. Liar Machine on Z Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=1

9. Liar Machine on Z Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=2

10. Liar Machine on Z Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=3

11. Liar Machine on Z Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=4

12. Liar Machine on Z Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=5

13. Liar Machine on Z Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=6

14. Liar Machine on Z Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 Height of linear machine at t=7 l 1 -distance: 5.80 l ∞ -distance: 0.98 t=7

15. Discrepancy for Two Discretizations Liar machine: round-offs spatially balanced Rotor-router model/Propp machine: round-offs temporally balanced The liar machine has poorer discrepancy… but provides bounds to a certain liar game.

16. Proof of Liar Machine Pointwise Discrepancy

17. The Liar Game A priori: M=#chips, n=#rounds, e=max #lies Initial configuration: f 0 = M ¢  0 Each round, obtain f t+1 from f t by: (1) Paul 2-colors the chips (2) Carole moves one color class left, the other right Final configuration: f n Winning conditions Original variant (Berlekamp, Rényi, Ulam) Pathological variant (Ellis, Yan)

18. Pathological Liar Game Bounds Fix n, e. Define M * (n,e) = minimum M such that Paul can win the pathological liar game with parameters M,n,e. Sphere Bound (E,P,Y `05) For fixed e, M * (n,e) · sphere bound + C e (Delsarte,Piret `86) For e/n 2 (0,1/2), M * (n,e) · sphere bound ¢ n ln 2. (C,E `09+) For e/n 2 (0,1/2), using the liar machine, M * (n,e) = sphere bound ¢.

19. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 19 9-9-8-7-6-5-4-3-2012345678 9 chips t=0 disqualified

20. 9-9-8-7-6-5-4-3-2012345678 20 9-9-8-7-6-5-4-3-2012345678 t=1 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

21. 9-9-8-7-6-5-4-3-2012345678 21 9-9-8-7-6-5-4-3-2012345678 t=2 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

22. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 22 9-9-8-7-6-5-4-3-2012345678 t=3 disqualified

23. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 23 9-9-8-7-6-5-4-3-2012345678 t=4 disqualified

24. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 24 9-9-8-7-6-5-4-3-2012345678 t=5 disqualified

25. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 25 9-9-8-7-6-5-4-3-2012345678 t=6 disqualified No chips survive: Paul loses

26. Comparison of Processes 26 ProcessOptimal #chips Linear machine9 1/7 (6,1)-Pathological liar game10 (6,1)-Liar machine12 9-9-8-7-6-5-4-3-2012345678 (6,1)-Liar machine started with 12 chips after 6 rounds disqualified

27. 9-9-8-7-6-5-4-3-2012345678 Loss from Liar Machine Reduction 27 9-9-8-7-6-5-4-3-2012345678 t=3 disqualified 9-9-8-7-6-5-4-3-2012345678 disqualified Paul’s optimal 2-coloring:

28. Reduction to Liar Machine

29. Saving One Chip in the Liar Machine 29

30. Summary: Pathological Liar Game Theorem

31. Further Exploration  Tighten the discrepancy analysis for the special case of initial chip configuration f 0 =M  0.  Generalize from binary questions to q-ary questions, q ¸ 2.  Improve analysis of the original liar game from Spencer and Winkler `92; solve the optimal rate of q-ary adaptive block codes for all fractional error rates.  Prove general pointwise and interval discrepancy theorems for various discretizations of random walks. 31

32. Reading List  This paper: Linearly bounded liars, adaptive covering codes, and deterministic random walks, preprint (see homepage).  The liar machine –Joel Spencer and Peter Winkler. Three thresholds for a liar. Combin. Probab. Comput.,1(1):81-93, 1992.  The pathological liar game –Robert Ellis, Vadim Ponomarenko, and Catherine Yan. The Renyi-Ulam pathological liar game with a fixed number of lies. J. Combin. Theory Ser. A, 112(2):328-336, 2005.  Discrepancy of deterministic random walks –Joshua Cooper and Joel Spencer, Simulating a Random Walk with Constant Error, Combinatorics, Probability, and Computing, 15 (2006), no. 06, 815-822. –Joshua Cooper, Benjamin Doerr, Joel Spencer, and Gabor Tardos. Deterministic random walks on the integers. European J. Combin., 28(8):2072-2090, 2007. 32