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 22nd, 2010Peled 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

6th cousins once removed?

Peled number <= 4:Peled -> Harary -> Erdős -> Chung -> Ellis

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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

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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

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Linear Machine on Z

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Linear Machine on Z

5.5 5.5

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Linear Machine on Z

2.75 5.5 2.75

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

Liar Machine on Z

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Liar machine time-stepNumber 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-stepNumber 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 -2 -1 0 1 2 3 4 5 6 7 8

t=1

Liar Machine on Z

Liar machine time-stepNumber 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 -2 -1 0 1 2 3 4 5 6 7 8

t=2

Liar Machine on Z

Liar machine time-stepNumber 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 -2 -1 0 1 2 3 4 5 6 7 8

t=3

Liar Machine on Z

Liar machine time-stepNumber 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 -2 -1 0 1 2 3 4 5 6 7 8

t=4

Liar Machine on Z

Liar machine time-stepNumber 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 -2 -1 0 1 2 3 4 5 6 7 8

t=5

Liar Machine on Z

Liar machine time-stepNumber 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 -2 -1 0 1 2 3 4 5 6 7 8

t=6

Liar Machine on Z

Liar machine time-stepNumber 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 -2 -1 0 1 2 3 4 5 6 7 8

Height of linear machine at t=7l1-distance: 5.80l∞-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 #liesInitial 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 conditionsOriginal 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 + Ce

(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 ¢ .

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Liar Machine vs. (6,1)-Pathological Liar Game19

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9 chips

9 chips

t=0

disqualified

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20

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

t=1

disqualified

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

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

t=2

disqualified

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

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Liar Machine vs. (6,1)-Pathological Liar Game22

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

t=3

disqualified

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Liar Machine vs. (6,1)-Pathological Liar Game23

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

t=4

disqualified

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Liar Machine vs. (6,1)-Pathological Liar Game24

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

t=5

disqualified

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Liar Machine vs. (6,1)-Pathological Liar Game25

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

t=6

disqualified

No chips survive: Paul loses

Comparison of Processes26

Process Optimal #chips

Linear machine 9 1/7

(6,1)-Pathological liar game 10

(6,1)-Liar machine 12

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(6,1)-Liar machine started with 12 chips after 6 rounds

disqualified

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Loss from Liar Machine Reduction27

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

disqualified

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8disqualified

Paul’s optimal 2-coloring:

Reduction to Liar Machine

Saving One Chip in the Liar Machine29

Summary: Pathological Liar Game Theorem

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.

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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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