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Identification of strategies for liar-type games via
discrepancy from their linear approximations
Robert Ellis
October 14th, 2011
AMS Sectional Meeting, Lincoln
Joint with Joshua Cooper, Daniel
Tietzer, Ruoran Wang, and James Williamson
Outline of Talk
 Diffusion processes on Z
– Simple random walk (linear machine)
– Liar games, and the pathological variant
– Liar machine
 Improved pathological liar game bound
– Reduction to liar machine
– Discrepancy analysis of liar machine versus linear machine
– Sub-optimality of the liar machine for the original liar game
 Concluding remarks
– Q-ary versions and group-testing versions
2
3
Linear Machine on Z
M = 11
11
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
g0 (initial configuration)
6
7
8
9
4
Linear Machine on Z
5.5
-9
-8
-7
-6
-5
-4
-3
-2
-1
5.5
0
1
2
g1 (t = 1)
3
4
5
6
7
8
9
5
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
g2 (t = 2)
Time-evolution of gt : M £ centered binomial
distribution of t {-1,+1} coin flips
6
7
8
9
The Liar Game, Encoded on Z
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
Chips to right of posn. –t + 2e ft in are eliminated.
Final configuration: fn
Liar game winning conditions
Original variant (Berlekamp, Rényi, Ulam)
Pathological variant (Ellis, Yan)
6
7
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 `10) For e/n 2 (0,1/2), using the liar machine,
M*(n,e) = sphere bound ¢
.
8
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
•Approximates linear machine
•Preserves indivisibility of chips
5
6
7
8
9
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
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
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
11
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
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=4
-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=5
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
14
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
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=7
-9
-8
-7
-6
-5
-4
-3
-2
-1
Height of linear machine at t=7
l1-distance: 5.80
l∞-distance: 0.98
0
1
2
3
4
5
6
7
8
9
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 encodes the
odds-vs.-evens question strategy for the liar game when Carole
always moves odd-numbered chips (optimal for her).
16
Proof of Liar Machine Pointwise Discrepancy
17
18
Liar Machine vs. (6,1)-Pathological Liar Game
9 chips
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
5
6
7
disqualified
8
9
t=0
9 chips
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
19
Liar Machine vs. (6,1)-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
6
disqualified
20
Liar Machine vs. (6,1)-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
5
disqualified
6
21
Liar Machine vs. (6,1)-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
4
disqualified
5
6
22
Liar Machine vs. (6,1)-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
3
disqualified
4
5
6
23
Liar Machine vs. (6,1)-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
2
disqualified
3
4
5
6
24
Liar Machine vs. (6,1)-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
1
disqualified
2
3
4
5
6
Liar Machine reduces to Pathological Game
Theorem (C,E `10). If
for the liar
machine, then Paul can win the pathological liar game with
the same initial configuration f0.
Proof ingredients.
 Put the weak majorization partial order on all chip
configurations with M chips (idea extended from
Spencer,Winkler `92)
 Carole maximizes the configuration in the order by always
moving the odd chips, thereby maximizing position of 1st chip
 The liar machine always moves the odd-numbered chips
25
26
Saving One Chip in the Liar Machine
n1 rounds
n2 rounds
Summary: Pathological Liar Game Theorem
27
Liar Machine for the Original Liar Game?
A priori: M=#chips, n=#rounds, e=max #lies
K’(n,e) = min M s.t. Paul can win the pathological liar game
K*(n,e) = min M s.t. liar machine preserves ≥ 1 chip
P’(n,e) = max M s.t. Paul can win the original liar game
P*(n,e) = max M s.t. move-evens liar machine preserves ≤ 1 chip
(Spencer,Winkler `86) If Paul asks odds-vs.-evens questions, Carole’s best
response is to move evens, encoded by the move-evens liar machine.
Question: Does the move-evens liar machine provide an asymptotically
good strategy for Paul in the original liar game?
Answer: No, suboptimal questioning strategy
28
Log Asymptotics of
29
P*(n,e)
(Pathological game, liar machine)
1
K’(f) := limn->∞ (1/n)log2K’(n,fn)
K*(f) := limn->∞ (1/n)log2K*(n,fn)
K*,K’
(Original game, move-evens machine)
P*
P’(f) := limn->∞ (1/n)log2
P*(f) := limn->∞ (1/n)log2P*(n,fn)
P’(n,fn)
P’
Theorem (Delsarte,Piret).
K*(f) = 1-h(f), where
h(f) = -f log2 f – (1-f) log2(1-f)
0
Theorem (E,Wang`10).
P*(n,e) ≤ K*(n-e,e)
0
f
1/3
(Berlekamp,Zigangirov) P’(f) = K*(f) until f=1/(3+51/2), then linear until f=1/3.
Q-ary Extensions of the Liar Machine/Pathological Game
Q-ary linear machine
Send (q-1)/q fraction right, 1/q fraction left; each posn.&round
Q-ary liar machine
(1) Number chips left-to-right 0,1,2,… take mod q of numbers
(2) Move classes 0,…,q-2 to right, class q-1 to left.
Q-ary liar game
(1) Paul partitions [M] into q parts.
(2) Carole picks one part and adds a lie to every element of the
other (q-1) parts
(E,T,W`11) Same orders for pointwise and interval maximum
discrepancy for q-ary case (different constants)
Paul has a winning strategy for M ≤ O( (ln ln n)1/2 * sphere bnd)
30
Q-ary Extensions of the Liar Machine/Pathological Game
Q-ary a-pooled linear machine
Send (q-a)/q fraction right, a/q fraction left; each posn.&round
Q-ary liar machine
(1) Number chips left-to-right 0,1,2,… take mod q of numbers
(2) Move classes 0,…,q-a-1 to right, classes q-a,…,q-1 to left.
Q-ary liar game
(1) Paul partitions [M] into q parts.
(2) Carole picks a parts and adds a lie to every element of the
other (q-a) parts
Group-testing: a positives in a group of M elements…
(E,T,W`11) Again, discrepancies and bound on M work out.
31
Further Exploration
 Solve the q-ary original liar game optimal number of chips for
all error rates using the liar machine framework as one step
 Analyze other group-testing models
 Convert winning strategies to a small number of batches
(adaptive -> nonadaptive strategies)
Thank you to the organizers. Questions?
32
Additional slides
Additional slides
Additional slides
36
Comparison of Processes
Process
Optimal #chips
Linear machine
9 1/7
(6,1)-Pathological liar game
10
(6,1)-Liar machine
12
-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
37
Loss from Liar Machine Reduction
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
t=3
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
disqualified
5
6
7
8
9
-1
0
1
2
3
4
disqualified
5
6
7
8
9
Paul’s optimal 2-coloring:
-9
-8
-7
-6
-5
-4
-3
-2
Reduction to Liar Machine
Outline of Talk
 Coding theory overview
– Packing (error-correcting) & covering codes
– Coding as a 2-player game
– Liar game and pathological liar game
 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
39
Coding Theory Overview
 Codewords:
fixed-length strings from a finite alphabet
 Primary uses:
Error-correction for transmission in the presence of noise
Compression of data with or without loss
 Viewpoints:
Packings and coverings of Hamming balls in the hypercube
2-player perfect information games
 Applications:
Cell phones, compact disks, deep-space communication
40
41
Coding Theory: (n,e)-Codes
 Transmit blocks of length n
 Noise changes
≤ e bits per block
(||||1 ≤ e)
x1…xn

(x1+1)…(xn+ n)
Richard Hamming
 Repetition code 111, 000
– length: n = 3
– e=1
– information rate: 1/3
Received:
110 010 000 101
blockwise majority vote
Decoded:
111 000 000 111
42
Coding Theory – A Hamming (7,1)-Code
Length n=7, corrects e=1 error
received
1 1 0 1 0 0 1
1 0
1 0 1
0 0
1 0 1
0
decoded
3 error:
errors:correct
incorrect
decoding
1
decoding
1 0 0 0 1 1 1
0 1 1 0 1 1 0
0 1 0 0 0 1 1
0 1 0 1 1 0 1
0 0 1 0 1 0 1
0 0 1 1 0 1 1
0 0 0 1 1 1 0
1 1 1 0 0 0 1
0 0 0 0 0 0 0
1 1 0 1 0 1 0
1 1 0 0 1 0 0
1 0 1 1 1 0 0
1 0 1 0 0 1 0
0 1 1 1 0 0 0
1 0 0 1 0 0 1
1 1 1 1 1 1 1
43
A Repetition Code as a Packing
 (3,1)-code:
111, 000
 Pairwise distance = 3 
1 error can be corrected
 The M codewords of an
(n,e)-code correspond to
a packing of Hamming balls
of radius e in the n-cube
111
110
101
011
100
010
001
000
A packing of 2
radius-1 Hamming balls
in the 3-cube
44
A (5,1)-Packing Code as a 2-Player Game
 (5,1)-code: 11111, 10100, 01010, 00001
Paul
Carole
11111
What is the 1st bit?
0
10100
What is the 2nd bit?
0
01010
What is the
3rd
bit?
0
What is the 4th bit?
1
What is the 5th bit?
0
00001
0
1
>1
# errors
11111
10100
01010
00001
01111
00100
00010
00011
45
Covering Codes
 Covering is the companion problem to packing
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
46
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
47
A (5,1)-Covering Code as a Football Pool
Round 1
Round 2
Round 3
Round 4
Round 5
Bet 1
W
W
W
W
W
11111
Bet 2
L
W
W
W
W
01111
Bet 3
W
L
W
W
W
10111
Bet 4
W
W
L
L
L
11000
Bet 5
L
L
W
L
L
00100
Bet 6
L
L
L
W
L
00010
Bet 7
L
L
L
L
W
00001
Payoff:
a bet with ≤ 1 bad prediction
Question. Min # bets to guarantee a payoff?
Ans.=7
48
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
A (5,1)-Adaptive Packing Code as a 2-Player Liar Game
Paul
Carole
A
Is the message C or D?
Y
Is the message A or C?
N
C
Is the message B?
N
D
Is the message D?
N
0
Is the message C?
Y
B
1
# lies
>1
Message
A
B
C
D
Original
encoding
00101
01110
01010
11000
10011
Adapted
encoding
1****
11***
101**
1****
10***
100**
10***
10001
1000*
1000*
10000
Y $ 1, N $ 0
49
A (5,1)-Adaptive Covering Code as a Football Pool
Round 1 Round 2 Round 3 Round 4 Round 5
Bet 1
W
W
Bet 1
W
Bet 2
Bet 2
W
L
W
W
L
Bet 3
Bet 3
W
L
L
L
W
Bet 4
Bet 4
L
W
Bet 6
Bet 5
L
W
0
Bet 6
L
W
Carole
W
L
L
W
Bet 5
L
Payoff:
a bet with ≤ 1 bad prediction
Question. Min # bets to guarantee a payoff?
1
>1
# bad
predictions
(# lies)
Ans.=6
50
51
Optimal (5,1)-Codes
Code type
Optimal size
(5,1)-code
4
(5,1)-adaptive code
4
Sphere bound
5 1/3 (= 25/(5+1) )
(5,1)-adaptive covering code
6
(5,1)-covering code
7
Adaptive Codes: Results and Questions
Sizes of optimal adaptive packing codes
•Binary, fixed e
≥ sphere bound - ce (Spencer `92)
•Binary, e=1,2,3 =sphere bound - O(1), exact solutions (Pelc; Guzicki; Deppe)
•Q-ary, e=1 =sphere bound - c(q,e), exact solution (Aigner `96)
•Q-ary, e linear unknown if rate meets Hamming bound for all e. (Ahlswede,
C. Deppe, and V. Lebedev)
Sizes of optimal adaptive covering codes
•Binary, fixed e · sphere bound + Ce
Binary, e=1,2 =sphere bound + O(1), exact solution (Ellis, Ponomarenko, Yan `05)
Near-perfect adaptive codes
•Q-ary, symmetric or “balanced”, e=1 exact solution (Ellis `04+)
•General channel, fixed e asymptotic first term (Ellis, Nyman `09)
52