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

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

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)

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)

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)

6

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 ¢ .

7

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

8

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

9

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

10

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

11

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

12

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

13

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

14

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

15

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 Discrepancy17

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

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

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

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

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

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

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

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

Saving One Chip in the Liar Machine26

n1 rounds

n2 rounds

Summary: Pathological Liar Game Theorem27

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

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

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

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

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:

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 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

40

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

41

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

42

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

43

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

44

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

45

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:

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

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

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

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

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

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)