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

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

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

Linear Machine on Z

Linear Machine on Z 5.5

Linear Machine on Z Time-evolution: 11 £ binomial distribution of {-1,+1} coin flips

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) 11 chips t=0 Approximates linear machine Preserves indivisibility of chips

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

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

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

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

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

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

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) Height of linear machine at t=7 l 1 -distance: 5.80 l ∞ -distance: 0.98 t=7

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.

Proof of Liar Machine Pointwise Discrepancy

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)

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 ¢.

Liar Machine vs. (6,1)-Pathological Liar Game chips t=0 disqualified

t=1 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

t=2 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

Liar Machine vs. (6,1)-Pathological Liar Game t=3 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=4 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=5 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=6 disqualified No chips survive: Paul loses

Comparison of Processes 26 ProcessOptimal #chips Linear machine9 1/7 (6,1)-Pathological liar game10 (6,1)-Liar machine (6,1)-Liar machine started with 12 chips after 6 rounds disqualified

Loss from Liar Machine Reduction t=3 disqualified disqualified Paul’s optimal 2-coloring:

Reduction to Liar Machine

Saving One Chip in the Liar Machine 29

Summary: Pathological Liar Game Theorem

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

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,  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): ,  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, –Joshua Cooper, Benjamin Doerr, Joel Spencer, and Gabor Tardos. Deterministic random walks on the integers. European J. Combin., 28(8): ,