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An Improved Liar Game Strategy From a Deterministic Random Walk Robert Ellis February 22 nd , 2010

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.

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An Improved Liar Game Strategy From a Deterministic Random Walk Robert Ellis February 22 nd , 2010

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

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

  3. 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

  4. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 Linear Machine on Z 11

  5. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 Linear Machine on Z 5.5 5.5

  6. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 Linear Machine on Z 2.75 5.5 2.75 Time-evolution: 11 £ binomial distribution of {-1,+1} coin flips

  7. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=0 11 chips • Approximates linear machine • Preserves indivisibility of chips

  8. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=1

  9. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=2

  10. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=3

  11. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=4

  12. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=5

  13. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=6

  14. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 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) t=7 Height of linear machine at t=7 l1-distance: 5.80 l∞-distance: 0.98

  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: f0 = M¢0 Each round, obtain ft+1 from ft by: (1) Paul 2-colors the chips (2) Carole moves one color class left, the other right Final configuration: fn 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 + Ce (Delsarte,Piret `86) For e/n2 (0,1/2), M*(n,e) · sphere bound ¢nln 2 . (C,E `09+) For e/n2 (0,1/2), using the liar machine, M*(n,e) = sphere bound ¢ .

  19. 19 Liar Machine vs. (6,1)-Pathological Liar Game 9 chips -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=0 9 chips disqualified

  20. 20 Liar Machine vs. (6,1)-Pathological Liar Game -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=1 disqualified

  21. 21 Liar Machine vs. (6,1)-Pathological Liar Game -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=2 disqualified

  22. 22 Liar Machine vs. (6,1)-Pathological Liar Game -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=3 disqualified

  23. 23 Liar Machine vs. (6,1)-Pathological Liar Game -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=4 disqualified

  24. 24 Liar Machine vs. (6,1)-Pathological Liar Game -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=5 disqualified

  25. 25 Liar Machine vs. (6,1)-Pathological Liar Game -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=6 No chips survive: Paul loses disqualified

  26. 26 Comparison of Processes -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 disqualified (6,1)-Liar machine started with 12 chips after 6 rounds

  27. 27 Loss from Liar Machine Reduction t=3 -9 -9 -9 -8 -8 -8 -7 -7 -7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 Paul’s optimal 2-coloring: disqualified disqualified

  28. Reduction to Liar Machine

  29. 29 Saving One Chip in the Liar Machine

  30. Summary: Pathological Liar Game Theorem

  31. 31 Further Exploration • Tighten the discrepancy analysis for the special case of initial chip configuration f0=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.

  32. 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.

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