An Improved Liar Game Strategy From a Deterministic Random Walk Robert Ellis February 22 nd , 2010
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Transcript of 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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Linear Machine on Z
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Linear Machine on Z
5.5 5.5
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Linear Machine on Z
2.75 5.5 2.75
Time-evolution: 11 £ binomial distribution of {-1,+1} coin flips
Liar Machine on Z
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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)
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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)
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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)
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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)
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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)
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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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t=1
disqualified
Liar Machine vs. (6,1)-Pathological Liar Game
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t=2
disqualified
Liar Machine vs. (6,1)-Pathological Liar Game
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Liar Machine vs. (6,1)-Pathological Liar Game22
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t=3
disqualified
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Liar Machine vs. (6,1)-Pathological Liar Game23
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t=4
disqualified
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Liar Machine vs. (6,1)-Pathological Liar Game24
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t=5
disqualified
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Liar Machine vs. (6,1)-Pathological Liar Game25
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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
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disqualified
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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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