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An Improved Liar Game Strategy From a
Deterministic Random Walk
Robert Ellis
Midwest Theory Day
December 5, 2009
Joint work with Joshua Cooper,
University of South Carolina
Outline of Talk
 Quick coding theory overview
 Diffusion processes on Z
– Simple random walk (linear machine)
– Liar machine
– Pathological liar game, alternating question strategy
 Improved pathological liar game bound
– Reduction to liar machine
– Discrepancy analysis of liar machine versus linear machine
 Concluding remarks
2
3
Coding Theory: (n,e)-Codes
 Transmit blocks of length n
 Noise changes ||||1 ≤ e bits per block
x1…xn
(x1+1)…(xn+ n)
receiver
sender
Richard Hamming
Noise 
Received:
 Repetition code 111, 000
length: n = 3, e = 1
110 010 000 101
blockwise majority vote
Decoded:
111 000 000 111
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Packing and Covering Codes
 Block codes ! packing and covering problems
111
 Packing: (n,e)-code
packing radius
length
110
101
011
100
010
001
covering radius
000
 Covering: (n,R)-code
11111
11111
00100
10100
01111
00010
01010
10111
00001
00001
11000
(5,1)-packing code
(5,1)-covering code
(3,1)-packing code and
(3,1)-covering code
“perfect code”
Optimal Length 5 Packing & Covering Codes
(5,1)-packing code
11111
11110 11101 11011 10111 01111
11100 11010 11001 10110 10101 10011 01110 01101 01011 00111
11000 10100 10010 10001 01100 01010 01001 00110 00101 00011
10000 01000 00100 00010 00001
11111
(5,1)-covering code
00000
11110 11101 11011 10111 01111
11100 11010 11001 10110 10101 10011 01110 01101 01011 00111
Sphere bound:
11000 10100 10010 10001 01100 01010 01001 00110 00101 00011
10000 01000 00100 00010 00001
00000
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Codes with Feedback (Adaptive Codes)
 Feedback
Noiseless, delay-less report of actual received bits
 Improves the number of decodable messages
E.g., from 20 to 28 messages for an (8,1)-code
1, 0, 1, 1, 0
1, 1, 1, 1, 0
receiver
sender
Noise
1, 1, 1, 1, 0
Noiseless Feedback
Elwyn Berlekamp
Packing and Covering Adaptive Codes
 Codes with feedback also have packing (liar game) and
covering (pathological liar game) variants
 See http://math.iit.edu/~rellis/talks/13liarWalk/LiarWalkIITAM.pptx
for full details:
–
–
–
–
A packing code as a 2-player game
A covering code as a football (betting) pool
An adaptive packing code as a 2-player liar game
An adaptive covering codes as a football (adaptive betting) pool
and as a 2-player pathological liar game
Packing/covering structures $ game strategies
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Optimal (5,1)-Codes
Code type
Optimal size
(5,1)-code
4
(5,1)-adaptive code
4
Sphere bound
5 1/3
(5,1)-adaptive covering code
6
(5,1)-covering code
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Linear Machine on Z
11
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Linear Machine on Z
5.5
-9
-8
-7
-6
-5
-4
-3
-2
-1
5.5
0
1
2
3
4
5
6
7
8
9
11
Linear Machine on Z
2.75
-9
-8
-7
-6
-5
-4
-3
-2
5.5
-1
0
2.75
1
2
3
4
5
Time-evolution is proportional to rows of Pascal’s triangle
6
7
8
9
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)
t=0
11 chips
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
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)
t=1
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
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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
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
15
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
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
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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
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
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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
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
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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
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
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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
-9
-8
-7
-6
-5
-4
-3
-2
-1
Height of linear machine at t=7
0
1
2
3
4
5
6
7
8
9
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(6,1)-Liar Game
Liar game time step
Paul bipartitions chips: green, purple
Carole moves one color to right
Paul’s goal: disqualify all but ≤1 chip after t=6 time steps
Paul bipartitions
Carole
moves purple
t=0
9 chips
0
1
2
disqualified
21
(6,1)-Liar Game
Liar game time step
Paul bipartitions chips: green, purple
Carole moves one color to right
Paul’s goal: disqualify all but ≤1 chip after t=6 time steps
t=1
Paul bipartitions
Carole
moves green
0
1
2
disqualified
22
(6,1)-Liar Game
Liar game time step
Paul bipartitions chips: green, purple
Carole moves one color to right
Paul’s goal: disqualify all but ≤1 chip after t=6 time steps
t=2
Paul bipartitions
Carole
moves green
0
1
2
disqualified
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(6,1)-Liar Game
Liar game time step
Paul bipartitions chips: green, purple
Carole moves one color to right
Paul’s goal: disqualify all but ≤1 chip after t=6 time steps
t=3
Paul bipartitions
Carole
moves purple
0
1
2
disqualified
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(6,1)-Liar Game
Liar game time step
Paul bipartitions chips: green, purple
Carole moves one color to right
Paul’s goal: disqualify all but ≤1 chip after t=6 time steps
t=4
Paul bipartitions
Carole
moves purple
0
1
2
disqualified
25
(6,1)-Liar Game
Liar game time step
Paul bipartitions chips: green, purple
Carole moves one color to right
Paul’s goal: disqualify all but ≤1 chip after t=6 time steps
t=5
Paul bipartitions
Carole
moves green
0
1
2
disqualified
26
(6,1)-Liar Game
Liar game time step
Paul bipartitions chips: green, purple
Carole moves one color to right
Paul’s goal: disqualify all but ≤1 chip after t=6 time steps
t=6
0
1
2
disqualified
Two chips survive: Paul loses
A Liar Game Strategy for Carole
 Weight function for n rounds left; xi = #chips with i lies:
 Lemma (Berlekamp)
 Refined sphere bound
Liar game. Carole keeps half of weight every step.
Initial weight > 2n ) Final weight >1 ) Carole wins.
Pathological variant. Carole reduces half of weight every step.
Initial weight < 2n ) Final weight <1 ) Carole wins.
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(6,1)-Pathological Liar Game
Paul’s goal: preserve ¸ 1 chip after t=6 time steps
wt6-t(x)=wt6(x)=26-1
Paul bipartitions
Carole
moves green
t=0
9 chips
0
1
2
disqualified
29
(6,1)-Pathological Liar Game
Paul’s goal: preserve ¸ 1 chip after t=6 time steps
wt5(x)=25-3
t=1
Paul bipartitions
Carole
moves green
0
1
2
disqualified
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(6,1)-Pathological Liar Game
Paul’s goal: preserve ¸ 1 chip after t=6 time steps
wt4(x)=24-2
t=2
Paul bipartitions
Carole
moves green
0
1
2
disqualified
31
(6,1)-Pathological Liar Game
Paul’s goal: preserve ¸ 1 chip after t=6 time steps
wt3(x)=23-1
t=3
Paul bipartitions
Carole
moves purple
0
1
2
disqualified
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(6,1)-Pathological Liar Game
Paul’s goal: preserve ¸ 1 chip after t=6 time steps
wt2(x)=22-1
t=4
Carole
moves purple
Paul bipartitions
0
1
2
disqualified
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(6,1)-Pathological Liar Game
Paul’s goal: preserve ¸ 1 chip after t=6 time steps
wt1(x)=21-1
t=5
Carole
moves green
Paul
bipartitions
0
1
2
disqualified
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(6,1)-Pathological Liar Game
Paul’s goal: preserve ¸ 1 chip after t=6 time steps
wt0(x)=20-1<1
t=6
0
1
2
disqualified
No chips survive: Paul loses
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Optimal (6,1)-Codes
Code type
Optimal #chips
(6,1)-code
8
(6,1)-adaptive code
(Liar game)
Sphere bound
8
(6,1)-adaptive covering code
(Pathological liar game)
(6,1)-covering code
10
9 1/7
12
New Approach to the Pathological Liar Game
Spencer and Winkler (`92) reduced the liar game to the liar
machine, a discrete diffusion process on the integer line.
Ellis and Yan (`04) introduced the pathological liar game.
Cooper and Spencer (`06) use discrepancy analysis to compare
the Propp-machine to simple random walk on Zd.
Here: (1) We reduce the pathological liar game to the liar machine,
(2) Use discrepancy analysis to compare the liar machine to simple
random walk on Z, and thereby
(3) Improve the best known pathological liar game strategy when the
number of lies is a constant fraction of the number of rounds.
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Liar Machine vs. Pathological Liar Game
9 chips
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
t=0
9 chips
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
disqualified
38
Liar Machine vs. Pathological Liar Game
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
t=1
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
disqualified
6
39
Liar Machine vs. Pathological Liar Game
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
t=2
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
disqualified
5
6
40
Liar Machine vs. Pathological Liar Game
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
t=3
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
disqualified
4
5
6
41
Liar Machine vs. Pathological Liar Game
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
t=4
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
disqualified
3
4
5
6
42
Liar Machine vs. Pathological Liar Game
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
t=5
-9
-8
-7
-6
-5
-4
-3
-2
-1 0
1
disqualified
2
3
4
5
6
43
Liar Machine vs. Pathological Liar Game
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
t=6
No chips survive: Paul loses
-9
-8
-7
-6
-5
-4
-3
-2 -1 0
disqualified
1
2
3
4
5
6
(6,1)-Pathological Liar Game, Liar Machine
Code type
Optimal #chips
Sphere bound
9 1/7
(6,1)-adaptive covering code
(Pathological liar game)
(6,1)-liar machine
10
12
-8 -7 -6 -5 -4optimum:
-3 -2 -1 0 1 Minimum
2 3 4 5 6number
7 8 9 of
(6,1)-liar-9 machine
initial chips for ¸ 1 chip disqualified
to be at position · -4 when t=6
(6,1)-Liar machine started with 12 chips after 6 rounds
44
Reduction to Liar Machine
45
Reduction to Liar Machine
46
Liar Machine Versus Linear Machine
47
Saving One Chip in the Liar Machine
48
Pathological Liar Game Theorem
49
Further Exploration
 Tighten the discrepancy analysis for the special case of initial
chip configuration f0(z)=M 0(z).
 Generalize from binary questions to q-ary questions, q ¸ 2.
 Improve analysis of the original liar game from Spencer and
Winkler `92.
 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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