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Identification of strategies for liar-type games via discrepancy from their linear approximations

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. Outline of Talk. 2. Diffusion processes on Z

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Identification of strategies for liar-type games via discrepancy from their linear approximations

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  1. 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

  2. Outline of Talk 2 • 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

  3. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 3 Linear Machine on Z M = 11 11 g0 (initial configuration)

  4. Linear Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 5.5 5.5 g1 (t = 1)

  5. Linear Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 5 2.75 5.5 2.75 g2 (t = 2) Time-evolution of gt : M£ centered binomial distribution of t {-1,+1} coin flips

  6. The Liar Game, Encoded on Z 6 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 + 2eft in are eliminated. Final configuration: fn Liar game winning conditions Original variant (Berlekamp, Rényi, Ulam) Pathological variant (Ellis, Yan)

  7. Pathological Liar Game Bounds 7 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/n2 (0,1/2), M*(n,e) · sphere bound ¢n ln 2 . (C,E `10) For e/n2 (0,1/2), using the liar machine, M*(n,e) = sphere bound ¢ .

  8. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 8 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 • Approximates linear machine • Preserves indivisibility of chips

  9. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 9 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

  10. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 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

  11. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 11 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

  12. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 12 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

  13. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 13 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

  14. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 14 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

  15. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 15 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 Height of linear machine at t=7 l1-distance: 5.80 l∞-distance: 0.98

  16. Discrepancy for Two Discretizations 16 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).

  17. Proof of Liar Machine Pointwise Discrepancy 17

  18. Liar Machine vs. (6,1)-Pathological Liar Game 18 9 chips -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=0 9 chips disqualified

  19. Liar Machine vs. (6,1)-Pathological Liar Game 19 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=1 disqualified

  20. Liar Machine vs. (6,1)-Pathological Liar Game 20 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=2 disqualified

  21. Liar Machine vs. (6,1)-Pathological Liar Game 21 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=3 disqualified

  22. Liar Machine vs. (6,1)-Pathological Liar Game 22 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=4 disqualified

  23. Liar Machine vs. (6,1)-Pathological Liar Game 23 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=5 disqualified

  24. Liar Machine vs. (6,1)-Pathological Liar Game 24 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=6 No chips survive: Paul loses disqualified

  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 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

  26. Saving One Chip in the Liar Machine 26 n1 rounds n2 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) 1 K*,K’ P* P’ 0 1/3 0 f 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 log2f – (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.

  30. Q-ary Extensions of the Liar Machine/Pathological Game 30 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)

  31. Q-ary Extensions of the Liar Machine/Pathological Game 31 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.

  32. Further Exploration 32 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?

  33. Additional slides

  34. Additional slides

  35. Additional slides

  36. Comparison of Processes 36 -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 37 t=3 -9 -9 -9 -8 -8 -8 -7 -7 -7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 Paul’s optimal 2-coloring: disqualified disqualified

  38. Reduction to Liar Machine

  39. Outline of Talk 39 • 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

  40. Coding Theory Overview 40 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

  41. Coding Theory: (n,e)-Codes x1…xn  (x1+1)…(xn+ n) Received: 110 010 000 101 blockwise majority vote Decoded: 111 000 000 111 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 Richard Hamming

  42. Coding Theory – A Hamming (7,1)-Code 1 1 1 0 0 0 0 1 1 0 0 0 1 0 3 errors: incorrect decoding 1 error: correct decoding 42 Length n=7, corrects e=1 error received 1 1 0 1 0 0 1 decoded

  43. A Repetition Code as a Packing 111 110 101 011 100 010 001 000 43 111 110 101 011 100 010 001 000 A packing of 2 radius-1 Hamming balls in the 3-cube (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

  44. A (5,1)-Packing Code as a 2-Player Game 01111 00100 00010 00011 44 Paul Carole 11111 What is the 1st bit? 0 10100 01010 What is the 2nd bit? 0 00001 What is the 3rd bit? 0 0 1 >1 What is the 4th bit? 1 What is the 5th bit? 0 # errors 11111 11111 10100 01010 00001 01010 00001 00001 00001 10100 01010 00001 00100 00010 00010 00010 (5,1)-code: 11111, 10100, 01010, 00001

  45. Covering Codes 111 110 101 011 111 100 010 001 110 101 011 000 100 010 001 000 45 packing radius length covering radius (3,1)-packing code and(3,1)-covering code“perfect code” 11111 11111 00100 10100 01111 00010 01010 10111 00001 00001 11000 (5,1)-packing code (5,1)-covering code Covering is the companion problem to packing Packing: (n,e)-code Covering: (n,R)-code

  46. Optimal Length 5 Packing & Covering Codes 46 (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 (5,1)-covering code 11111 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 47 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) 48 1, 1, 1, 1, 0 1, 0, 1, 1, 0 receiver sender Noise 1, 1, 1, 1, 0 Elwyn Berlekamp Noiseless Feedback 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

  49. A (5,1)-Adaptive Packing Code as a 2-Player Liar Game 10*** 1**** 10*** 1000* 11*** 1**** 101** 100** 10001 10000 49 Paul Carole A Is the message C or D? Y B Is the message A or C? N C Is the message B? N D Is the message D? N 0 1 >1 Is the message C? Y # lies A B D Message C Originalencoding 00101 01110 01010 11000 10011 Adaptedencoding 1000* Y $ 1, N $ 0

  50. A (5,1)-Adaptive Covering Code as a Football Pool W W L W W L L L L W W W W 50 Round 1 Round 2 Round 3 Round 4 Round 5 Bet 1 W Bet 1 Bet 2 Bet 2 W Bet 3 Bet 3 W Bet 4 Bet 5 Bet 4 L Bet 6 Bet 5 L 0 1 >1 # badpredictions(# lies) Bet 6 L Carole W L L W L Payoff: a bet with ≤ 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6

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