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  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

Linear Machine on Z g 0 (initial configuration) M = 11

Linear Machine on Z g 1 (t = 1)

Linear Machine on Z Time-evolution of g t : M £ centered binomial distribution of t {-1,+1} coin flips 5 g 2 (t = 2)

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

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

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) 11 chips t=0 Approximates linear machine Preserves indivisibility of chips 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=1 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=2 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=3 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=4 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=5 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=6 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) Height of linear machine at t=7 l 1 -distance: 5.80 l ∞ -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 Discrepancy 17

Liar Machine vs. (6,1)-Pathological Liar Game chips t=0 disqualified

t=1 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

t=2 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

Liar Machine vs. (6,1)-Pathological Liar Game t=3 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=4 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=5 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=6 disqualified No chips survive: Paul loses

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

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

Summary: Pathological Liar Game Theorem 27

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

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

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

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

Loss from Liar Machine Reduction t=3 disqualified disqualified 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 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

 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 ) Received: Decoded: blockwise majority vote Richard Hamming 41

errors: incorrect decoding Coding Theory – A Hamming (7,1)-Code Length n=7, corrects e=1 error received decoded 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 to a packing of Hamming balls of radius e in the n -cube 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, What 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 >1 # errors

Covering Codes  Covering is the companion problem to packing  Packing: (n,e) -code  Covering: (n,R) -code length packing radius covering radius (3,1) -packing code and (3,1) -covering code “perfect code” (5,1)-packing code(5,1)-covering code 45

Optimal Length 5 Packing & Covering Codes (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 prediction Question. Min # bets to guarantee a payoff? Ans.=

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

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 Message Original encoding Adapted encoding A B C D **** 11*** 10*** 1000* 101**100** 1000* Y $ 1, N $ 0 49

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 >1 # bad predictions (# lies) Bet 2 Bet 1 50

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

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)