Presentation is loading. Please wait.

Presentation is loading. Please wait.

Identification of strategies for liar-type games via discrepancy from their linear approximations Robert Ellis October 14 th, 2011 AMS Sectional Meeting,

Published byBeatrix Pierce Modified over 9 years ago

Similar presentations

Presentation on theme: "Identification of strategies for liar-type games via discrepancy from their linear approximations Robert Ellis October 14 th, 2011 AMS Sectional Meeting,"— Presentation transcript:

1 Identification of strategies for liar-type games via discrepancy from their linear approximations Robert Ellis October 14 th, 2011 AMS Sectional Meeting, Lincoln Joint with Joshua Cooper, Daniel Tietzer, Ruoran Wang, and James Williamson

2 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 9-9-8-7-6-5-4-3-2012345678 3 11 Linear Machine on Z g 0 (initial configuration) M = 11

4 9-9-8-7-6-5-4-3-2012345678 Linear Machine on Z 5.5 4 g 1 (t = 1)

5 9-9-8-7-6-5-4-3-2012345678 Linear Machine on Z 2.755.52.75 Time-evolution of g t : M £ centered binomial distribution of t {-1,+1} coin flips 5 g 2 (t = 2)

6 The Liar Game, Encoded on Z A priori: M=#chips, n=#rounds, e=max #lies Initial configuration: f 0 = M ¢  0 Each round, obtain f t+1 from f t by: (1) Paul 2-colors the chips (2) Carole moves one color class left, the other right Chips to right of posn. –t + 2e f t in are eliminated. Final configuration: f n 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 + C e (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

8 Liar Machine on Z 9-9-8-7-6-5-4-3-2012345678 Liar machine time-step Number 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

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) 9-9-8-7-6-5-4-3-2012345678 t=1 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) 9-9-8-7-6-5-4-3-2012345678 t=2 10

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) 9-9-8-7-6-5-4-3-2012345678 t=3 11

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) 9-9-8-7-6-5-4-3-2012345678 t=4 12

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) 9-9-8-7-6-5-4-3-2012345678 t=5 13

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) 9-9-8-7-6-5-4-3-2012345678 t=6 14

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) 9-9-8-7-6-5-4-3-2012345678 Height of linear machine at t=7 l 1 -distance: 5.80 l ∞ -distance: 0.98 t=7 15

16 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

17 Proof of Liar Machine Pointwise Discrepancy 17

18 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 18 9-9-8-7-6-5-4-3-2012345678 9 chips t=0 disqualified

19 9-9-8-7-6-5-4-3-2012345678 19 9-9-8-7-6-5-4-3-2012345678 t=1 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

20 9-9-8-7-6-5-4-3-2012345678 20 9-9-8-7-6-5-4-3-2012345678 t=2 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

21 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 21 9-9-8-7-6-5-4-3-2012345678 t=3 disqualified

22 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 22 9-9-8-7-6-5-4-3-2012345678 t=4 disqualified

23 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 23 9-9-8-7-6-5-4-3-2012345678 t=5 disqualified

24 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. (6,1)-Pathological Liar Game 24 9-9-8-7-6-5-4-3-2012345678 t=6 disqualified No chips survive: Paul loses

25 Liar Machine reduces to Pathological Game 25 Theorem (C,E `10). If for the liar machine, then Paul can win the pathological liar game with the same initial configuration f 0. 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 1 st chip  The liar machine always moves the odd-numbered chips

26 Saving One Chip in the Liar Machine 26 n 1 rounds n 2 rounds

27 Summary: Pathological Liar Game Theorem 27

28 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 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

29 Log Asymptotics of P * (n,e) 29 (Pathological game, liar machine) K’(f) := lim n->∞ (1/n)log 2 K ’ (n,fn) K*(f) := lim n->∞ (1/n)log 2 K * (n,fn) (Original game, move-evens machine) P’(f) := lim n->∞ (1/n)log 2 P ’ (n,fn) P*(f) := lim n->∞ (1/n)log 2 P * (n,fn) Theorem (Delsarte,Piret). K*(f) = 1-h(f), where h(f) = -f log 2 f – (1-f) log 2 (1-f) Theorem (E,Wang`10). P*(n,e) ≤ K*(n-e,e) (Berlekamp,Zigangirov) P’(f) = K*(f) until f=1/(3+5 1/2 ), then linear until f=1/3. K *,K’ P*P* P’P’ 0 1/3 0 1 f

30 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

31 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

32 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

33 Additional slides

34

35

36 Comparison of Processes 36 ProcessOptimal #chips Linear machine9 1/7 (6,1)-Pathological liar game10 (6,1)-Liar machine12 9-9-8-7-6-5-4-3-2012345678 (6,1)-Liar machine started with 12 chips after 6 rounds disqualified

37 9-9-8-7-6-5-4-3-2012345678 Loss from Liar Machine Reduction 37 9-9-8-7-6-5-4-3-2012345678 t=3 disqualified 9-9-8-7-6-5-4-3-2012345678 disqualified Paul’s optimal 2-coloring:

38 Reduction to Liar Machine

39 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

40 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  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…xnx1…xn (x 1 +  1 )…(x n +  n ) 110010000 101 000111 Received: Decoded: blockwise majority vote Richard Hamming 41

42 0010011 3 errors: incorrect decoding Coding Theory – A Hamming (7,1)-Code 10001110110110 01000110101101 00101010011011 00011101110001 00000001101010 11001001011100 10100100111000 10010011111111 Length n=7, corrects e=1 error 1001011 received decoded 1001001 1 error: correct decoding 42

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 110011101 111 000 010001100 000 010001100 110011101 111 A packing of 2 radius-1 Hamming balls in the 3-cube 43

44 A (5,1) -Packing Code as a 2-Player Game  (5,1)-code: 11111, 10100, 01010, 00001 0What is the 5 th bit? 1What is the 4 th bit? 0What is the 3 rd bit? 0What is the 2 nd bit? 0What is the 1 st bit? CarolePaul 11111 00001 10100 01010 0 1 >1 # errors 11111000011010001010 01111001000001000011 00100 01010 00010 00001 11111 101000101000001 44

45 Covering Codes  Covering is the companion problem to packing  Packing: (n,e) -code  Covering: (n,R) -code length packing radius covering radius 110011101 111 000 010001100 000 010001100 110011101 111 (3,1) -packing code and (3,1) -covering code “perfect code” 11111 00001 10100 01010 11111 11000 01111 1011100001 00100 00010 (5,1)-packing code(5,1)-covering code 45

46 Optimal Length 5 Packing & Covering Codes 01001 01100 01110 01101 00100 11100 01000 111101110101111 00000 01010 11000 10100 00110 00101 10110 10011 10001 10010 11011 00011 10111 000010001010000 11111 10101 00111 01011 11001 11010 0111001101 0100101100 00100 11100 01000 111101110101111 00000 0101011000101000011000101 1011010011 1000110010 11011 00011 10111 000010001010000 11111 1010100111010111100111010 (5,1) -packing code (5,1) -covering code 46 Sphere bound:

47 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 prediction Question. Min # bets to guarantee a payoff? Ans.=7 00100 01111 11000 10111 00001 00010 11111 47

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 sender receiver Noise Noiseless Feedback Elwyn Berlekamp 1, 0, 1, 1, 0 1, 1, 1, 1, 0 48

49 A (5,1) -Adaptive Packing Code as a 2-Player Liar Game A D B C 0 1 >1 # lies YIs 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 Original encoding Adapted encoding A B C D 0111001010 11000 10011 1**** 11*** 10*** 1000* 101**100** 1000* 1000010001 Y $ 1, N $ 0 49

50 A (5,1)-Adaptive Covering Code as a Football Pool LWLLW Carole L Bet 6 L Bet 5 L Bet 4 W Bet 3 W L L WW Bet 2 L W W W W W L L WW Bet 1 Round 5Round 4Round 3Round 2Round 1 Payoff: a bet with ≤ 1 bad prediction Question. 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

51 Optimal (5,1)-Codes 51 Code typeOptimal size (5,1)-code4 (5,1)-adaptive code4 Sphere bound5 1/3 (= 2 5 /(5+1) ) (5,1)-adaptive covering code6 (5,1)-covering code7

52 Adaptive Codes: Results and Questions 52 Sizes of optimal adaptive packing codes Binary, fixed e ≥ sphere bound - c e (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 + C e 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)

Download ppt "Identification of strategies for liar-type games via discrepancy from their linear approximations Robert Ellis October 14 th, 2011 AMS Sectional Meeting,"

Similar presentations

Ulams Game and Universal Communications Using Feedback Ofer Shayevitz June 2006.

Ulams Game and Universal Communications Using Feedback Ofer Shayevitz June 2006.
A threshold of ln(n) for approximating set cover By Uriel Feige Lecturer: Ariel Procaccia.

A threshold of ln(n) for approximating set cover By Uriel Feige Lecturer: Ariel Procaccia.
Approximate List- Decoding and Hardness Amplification Valentine Kabanets (SFU) joint work with Russell Impagliazzo and Ragesh Jaiswal (UCSD)

Approximate List- Decoding and Hardness Amplification Valentine Kabanets (SFU) joint work with Russell Impagliazzo and Ragesh Jaiswal (UCSD)
Inapproximability of MAX-CUT Khot,Kindler,Mossel and O ’ Donnell Moshe Ben Nehemia June 05.

Inapproximability of MAX-CUT Khot,Kindler,Mossel and O ’ Donnell Moshe Ben Nehemia June 05.
Noise, Information Theory, and Entropy (cont.) CS414 – Spring 2007 By Karrie Karahalios, Roger Cheng, Brian Bailey.

Noise, Information Theory, and Entropy (cont.) CS414 – Spring 2007 By Karrie Karahalios, Roger Cheng, Brian Bailey.
Information theory Multi-user information theory A.J. Han Vinck Essen, 2004.

Information theory Multi-user information theory A.J. Han Vinck Essen, 2004.
Introduction to Computer Science 2 Lecture 7: Extended binary trees

Introduction to Computer Science 2 Lecture 7: Extended binary trees
Applied Algorithmics - week7

Applied Algorithmics - week7
Probabilistic verification Mario Szegedy, Rutgers www/cs.rutgers.edu/~szegedy/07540 Lecture 4.

Probabilistic verification Mario Szegedy, Rutgers www/cs.rutgers.edu/~szegedy/07540 Lecture 4.
Gillat Kol (IAS) joint work with Ran Raz (Weizmann + IAS) Interactive Channel Capacity.

Gillat Kol (IAS) joint work with Ran Raz (Weizmann + IAS) Interactive Channel Capacity.
“Ulam‘s” Liar Game with Lies in an Interval Benjamin Doerr (MPI Saarbrücken, Germany) joint work with Johannes Lengler and David Steurer (Universität des.

“Ulam‘s” Liar Game with Lies in an Interval Benjamin Doerr (MPI Saarbrücken, Germany) joint work with Johannes Lengler and David Steurer (Universität des.
MIT and James Orlin © 2003 1 Game Theory 2-person 0-sum (or constant sum) game theory 2-person game theory (e.g., prisoner’s dilemma)

MIT and James Orlin © Game Theory 2-person 0-sum (or constant sum) game theory 2-person game theory (e.g., prisoner’s dilemma)
Games of Prediction or Things get simpler as Yoav Freund Banter Inc.

Games of Prediction or Things get simpler as Yoav Freund Banter Inc.
Information Theory Introduction to Channel Coding Jalal Al Roumy.

Information Theory Introduction to Channel Coding Jalal Al Roumy.
Day 2 Information theory ( 信息論 ) Civil engineering ( 土木工程 ) Cultural exchange.

Day 2 Information theory ( 信息論 ) Civil engineering ( 土木工程 ) Cultural exchange.
Fundamental limits in Information Theory Chapter 10 :

Fundamental limits in Information Theory Chapter 10 :
Fountain Codes Amin Shokrollahi EPFL and Digital Fountain, Inc.

Fountain Codes Amin Shokrollahi EPFL and Digital Fountain, Inc.
Coding Theory: Packing, Covering, and 2-Player Games Robert Ellis Menger Day 2008: Recent Applied Mathematics Research Advances April 14, 2008.

Coding Theory: Packing, Covering, and 2-Player Games Robert Ellis Menger Day 2008: Recent Applied Mathematics Research Advances April 14, 2008.
Spatial and Temporal Data Mining

Spatial and Temporal Data Mining
Probabilistic Methods in Coding Theory: Asymmetric Covering Codes Joshua N. Cooper UCSD Dept. of Mathematics Robert B. Ellis Texas A&M Dept. of Mathematics.

Probabilistic Methods in Coding Theory: Asymmetric Covering Codes Joshua N. Cooper UCSD Dept. of Mathematics Robert B. Ellis Texas A&M Dept. of Mathematics.

Similar presentations

Ads by Google