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![Page 1: Identification of strategies for liar-type games via discrepancy from their linear approximations Robert Ellis October 14 th, 2011 AMS Sectional Meeting,](https://reader036.fdocuments.net/reader036/viewer/2022062421/56649e055503460f94af1b22/html5/thumbnails/1.jpg)
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
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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
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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
3
11
Linear Machine on Z
g0 (initial configuration)
M = 11
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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
4
g1 (t = 1)
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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 of gt : M £ centered binomial distribution of t {-1,+1} coin flips
5
g2 (t = 2)
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The Liar Game, Encoded on Z
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 rightChips to right of posn. –t + 2e ft in are eliminated.
Final configuration: fn
Liar game winning conditionsOriginal variant (Berlekamp, Rényi, Ulam)
Pathological variant (Ellis, Yan)
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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 ¢ .
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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
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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
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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
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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
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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
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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
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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
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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
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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).
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Proof of Liar Machine Pointwise Discrepancy17
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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 Game18
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
9 chips
9 chips
t=0
disqualified
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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
19
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
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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
20
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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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 Game21
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=3
disqualified
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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 Game22
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=4
disqualified
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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=5
disqualified
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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=6
disqualified
No chips survive: Paul loses
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Liar Machine reduces to Pathological Game25
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
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Saving One Chip in the Liar Machine26
n1 rounds
n2 rounds
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Summary: Pathological Liar Game Theorem27
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Liar Machine for the Original Liar Game?28
A priori: M=#chips, n=#rounds, e=max #lies
K’(n,e) = min M s.t. Paul can win the pathological liar gameK*(n,e) = min M s.t. liar machine preserves ≥ 1 chip
P’(n,e) = max M s.t. Paul can win the original liar gameP*(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
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Log Asymptotics of P*(n,e)29
(Pathological game, liar machine)K’(f) := limn->∞ (1/n)log2K’(n,fn)K*(f) := limn->∞ (1/n)log2K*(n,fn)
(Original game, move-evens machine)P’(f) := limn->∞ (1/n)log2P’(n,fn)P*(f) := limn->∞ (1/n)log2P*(n,fn)
Theorem (Delsarte,Piret). K*(f) = 1-h(f), where h(f) = -f log2 f – (1-f) log2(1-f)
Theorem (E,Wang`10).P*(n,e) ≤ K*(n-e,e)
(Berlekamp,Zigangirov) P’(f) = K*(f) until f=1/(3+51/2), then linear until f=1/3.
K*,K’
P*
P’
0 1/3
0
1
f
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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)
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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.
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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?
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Additional slides
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Additional slides
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Additional slides
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Comparison of Processes36
Process Optimal #chips
Linear machine 9 1/7
(6,1)-Pathological liar game 10
(6,1)-Liar machine 12
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
(6,1)-Liar machine started with 12 chips after 6 rounds
disqualified
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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Loss from Liar Machine Reduction37
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 8disqualified
Paul’s optimal 2-coloring:
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Reduction to Liar Machine
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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
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Coding Theory Overview
Codewords:fixed-length strings from a finite alphabet
Primary uses: Error-correction for transmission in the presence of noiseCompression of data with or without loss
Viewpoints:Packings and coverings of Hamming balls in the hypercube2-player perfect information games
Applications:Cell phones, compact disks, deep-space communication
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Transmit blocks of length n
Noise changes≤ e bits per block(||||1 ≤ e)
Repetition code 111, 000– length: n = 3 – e = 1– information rate: 1/3
Coding Theory: (n,e)-Codes
x1…xn
(x1+1)…(xn+ n)
110 010 000
000
101
000 111111
Received:
Decoded:
blockwise majority vote
Richard Hamming
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0010011
3 errors: incorrect decoding
Coding Theory – A Hamming (7,1)-Code
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
Length n=7, corrects e=1 error
1001011
received
decoded
1001001
1 error: correct decoding
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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 toa packing of Hamming ballsof radius e in the n-cube
110 011101
111
000
010 001100
000
010 001100
110 011101
111
A packing of 2 radius-1 Hamming balls
in the 3-cube
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A (5,1)-Packing Code as a 2-Player Game
(5,1)-code: 11111, 10100, 01010, 00001
0What is the 5th bit?
1What is the 4th bit?
0What is the 3rd bit?
0What is the 2nd bit?
0What is the 1st bit?
CarolePaul 11111
00001
10100
01010
0 1 >1
# errors
11111 0000110100 01010
01111 00100 00010 0001100100
01010
000100001000010
00001000010000111111 10100 01010 00001
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Covering Codes
Covering is the companion problem to packing
Packing: (n,e)-code
Covering: (n,R)-code
lengthpacking radius
covering radius
110 011101
111
000
010 001100
000
010 001100
110 011101
111
(3,1)-packing code and(3,1)-covering code
“perfect code”11111
00001
10100
01010
11111
11000
01111
10111 00001
00100
00010
(5,1)-packing code (5,1)-covering code
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Optimal Length 5 Packing & Covering Codes
0100101100
01110 01101
00100
11100
01000
11110 11101 01111
00000
0101011000 10100 00110 00101
10110 10011
1000110010
11011
00011
10111
000010001010000
11111
10101 00111010111100111010
01110 01101
0100101100
00100
11100
01000
11110 11101 01111
00000
0101011000 10100 00110 00101
10110 10011
1000110010
11011
00011
10111
000010001010000
11111
10101 00111010111100111010
(5,1)-packing code
(5,1)-covering code
46
Sphere bound:
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A (5,1)-Covering Code as a Football Pool
WLLLLBet 7
LWLLLBet 6
LLWLLBet 5
LLLWWBet 4
WWWLWBet 3
WWWWLBet 2
WWWWWBet 1
Round 5Round 4Round 3Round 2Round 1
Payoff: a bet with ≤ 1 bad predictionQuestion. Min # bets to guarantee a payoff? Ans.=7
00100
01111
11000
10111
00001
00010
11111
47
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Codes with Feedback (Adaptive Codes)
FeedbackNoiseless, delay-less report of actual received bits
Improves the number of decodable messagesE.g., from 20 to 28 messages for an (8,1)-code
sender receiver
Noise
Noiseless FeedbackElwyn Berlekamp
1, 0, 1, 1, 0 1, 1, 1, 1, 0
1, 1, 1, 1, 0
48
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A (5,1)-Adaptive Packing Code as a 2-Player Liar Game
A
D
B
C
0 1 >1# liesYIs the message C?
NIs the message D?
NIs the message B?
NIs the message A or C?
YIs the message C or D?
CarolePaul
00101
Message
Originalencoding
Adaptedencoding
A B C D
01110 0101011000 10011
1**** 1****11*** 10*** 10*** 1000*101** 100**1000* 1000010001
Y $ 1, N $ 0
49
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A (5,1)-Adaptive Covering Code as a Football Pool
LWLLWCarole
LBet 6
LBet 5
LBet 4
WBet 3 W
L
L
WWBet 2
L
W
W
W
W
W
L
L
WWBet 1
Round 5Round 4Round 3Round 2Round 1
Payoff: a bet with ≤ 1 bad predictionQuestion. Min # bets to guarantee a payoff?
Ans.=6
Bet 3
Bet 6
Bet 4
Bet 5
0 1 >1# bad
predictions(# lies)
Bet 2
Bet 1
50
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Optimal (5,1)-Codes51
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
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Adaptive Codes: Results and Questions52
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)