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“Ulam‘s” Liar Game with Lies in an Interval

“Ulam‘s” Liar Game with Lies in an Interval. Benjamin Doerr (MPI Saarbr ücken, Germany). joint work with Johannes Lengler and David Steurer (Universität des Saarlandes, Germany). ADFOCS. Advanced Course on the Foundations of Computer Science. August 21 - August 25, 2006, Saarbrücken, Germany.

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“Ulam‘s” Liar Game with Lies in an Interval

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  1. “Ulam‘s” Liar Game with Lies in an Interval Benjamin Doerr (MPI Saarbrücken, Germany) joint work with Johannes Lengler and David Steurer (Universität des Saarlandes, Germany)

  2. ADFOCS Advanced Course on the Foundations of Computer Science August 21 - August 25, 2006, Saarbrücken, Germany Tamal Dey Joel Spencer Ingo Wegener Surface Reconstruction and Meshing: Algorithms with Mathematical Analysis Erdős Magic,Erdős-Rényi Phase Transition Randomized Search Heuristics: Concept and Analysis Early registration deadline: July 21! Benjamin Doerr Liar Games with Lies in an Interval

  3. Overview • Introduction to Liar Games • Basic Problem • Motivation: Noisy Communication • History • New Game: Lies in an interval • Problem • Result • Example • Some proof details Benjamin Doerr Liar Games with Lies in an Interval

  4. Basic Problem: Liar Games • Start: Carole thinks of a number x from 1 to n. • q Rounds: • Paul asks a YES/NO question (“Is x in S?”). • Carole answers, possibly faulty (‘lie’). • End: Paul wins if he knows the number x. Benjamin Doerr Liar Games with Lies in an Interval

  5. Liar Games & Noisy Communication 0 0 0 Satellite Task: Satellite sends data to base station. 0 0 Problem: Transmission errors[noisy communication]. Solution: Allow two-way communication[reciever may confirm/ask particular data].Assume: No errors on back-wards channel. 1 1 0 0 0 Base station Benjamin Doerr Liar Games with Lies in an Interval

  6. Liar Games: Worst-Case View • Start: Carole does not yet decide on the number x. • q Rounds: • Paul asks a YES/NO question (“Is x in S?”). • Carole gives some answers. • End: Paul wins if • he knows the number (there is only one possible number left), • he knows that Carole was cheating (no possible number left). Perfect information game: Clear who wins (in theory) Benjamin Doerr Liar Games with Lies in an Interval

  7. ¡ ¢ = = ( ) q q q q 2 2 2 1 · · · + n n n q k · Liar Games: History • Problem: • Ulam (1976): “Adventures of a mathematician”. • Renyi (1961,1976): In Hungarian (overlooked by most of the community). • Cicalese, Vaccaro (1998/99): “Renyi-Ulam game”. • Results: • No lie: Paul wins if . [trivial] • One lie: Roughly, Paul wins if . [Pelc (1987)] • k lies: Roughly, Paul wins if . [Spencer (1992)] • ... [120 References in Pelc’s survey paper (2002)] Benjamin Doerr Renyi-Ulam Liar Games with Lies in an Interval

  8. New Game: Lies in an Interval • Rules of the game: • Carole may lie up to k times, but: • All lies have to be in an interval of k consecutive rounds. • Other rules: As before. • Motivation: Noisy Communication • One disorder occurs. • Takes k rounds. • No reliable communication within that period. Benjamin Doerr Liar Games with Lies in an Interval

  9. k k l l l l · ¸ d d e e d e l l l l k k k k l l l l 2 2 2 2 1 ¸ ¸ o o g g o o g g n n + + + + + + ¡ < < q q q q o o o o g g g g n n n n o g o o g g o n g n Lies in an Interval: Results • Paul wins if • Carole wins if • Two Cases • (left inequalities) • (right inequalities) and or Benjamin Doerr Liar Games with Lies in an Interval

  10. k l l · l l k l l l l k o g o g n + + + ¼ ¼ q q o o g g n n o g o g o g o n g n Interval vs. arbitrary lies ( ) • Interval of length k: • k arbitrary lies: • Interval of length k: Paul needs k more questions than for one lie. [as it should be] Critical value Critical value Benjamin Doerr Liar Games with Lies in an Interval

  11. Let’s Play! (n = 10 ‘secrets’, k = 2 lies) Start: All secrets 1, ..., 10 are possible. Round 1: P: “Is x in {1, ..., 5}? C: “Yes!” Result: 1, ..., 5: Possible 6, ..., 10: Possible, if lied this round Round 2: P: “Is x in {1, 2, 6, 7, 8}? C: “Yes!” Result: 1, 2: Possible 3, 4, 5: Possible, if lied this round only 6, ..., 10: Possible, if lied one round ago Round 3: P: “Is x in {1, 3, 6, 7}? C: “Yes!” Result: 1: Possible 2: Possible, if lied this round only 3, 4, 5: Possible, if lied one round ago only 6, 7: Possible, if lied two rounds ago 8, 9, 10: Not possible (lied in round 1 and 3) Benjamin Doerr Liar Games with Lies in an Interval

  12. Let’s Play! (n = 10 ‘secrets’, k = 2 lies) P = (10, 0, 0) Start: All secrets 1, ..., 10 are possible. P = (5, 5, 0) Round 1: P: “Is x in {1, ..., 5}? C: “Yes!” Result: 1, ..., 5: Possible 6, ..., 10: Possible, if lied this round Round 2: P: “Is x in {1, 2, 6, 7, 8}? C: “Yes!” Result: 1, 2: Possible 3, 4, 5: Possible, if lied this round only 6, ..., 10: Possible, if lied one round ago P = (2, 3, 5) P = (1, 1, 5) Round 3: P: “Is x in {1, 3, 6, 7}? C: “Yes!” Result: 1: Possible 2: Possible, if lied this round only 3, ..., 7: Possible, if lied 1+ rounds ago (no further lie possible) 8, 9, 10: Not possible (lied in round 1 and 3) Position P = (xk, ..., x0): xk = # of possible secrets with no lie xi = # of secrets with first lie k-i rounds ago x0 = # of secrets with no lies allowed Benjamin Doerr Liar Games with Lies in an Interval

  13. Let’s Play! (n = 10 ‘secrets’, k = 2 lies) P = (10, 0, 0) Start: All secrets 1, ..., 10 are possible. Q = (5,0,0) P = (5, 5, 0) Round 1: P: “Is x in {1, ..., 5}? C: “Yes!” Result: 1, ..., 5: Possible 6, ..., 10: Possible, if lied this round Q = (2,3,0) Round 2: P: “Is x in {1, 2, 6, 7, 8}? C: “Yes!” Result: 1, 2: Possible 3, 4, 5: Possible, if lied this round only 6, ..., 10: Possible, if lied one round ago P = (2, 3, 5) Q = (1,1,2) P = (1, 1, 5) Round 3: P: “Is x in {1, 3, 6, 7}? C: “Yes!” Result: 1: Possible 2: Possible, if lied this round only 3, ..., 7: Possible, if lied 1+ rounds ago (no further lie possible) 8, 9, 10: Not possible (lied in round 1 and 3) Position P = (xk, ..., x0): xk = # of possible secrets with no lie xi = # of secrets with first lie k-i rounds ago x0 = # of secrets with no lies allowed Question Q = (xk, ..., x0) Benjamin Doerr Liar Games with Lies in an Interval

  14. Rules in Vector Format • Start: P = (n, 0, ..., 0). • q Rounds: • P = (xk, xk-1, xk-2, ..., x1, x0) • Q = (yk, yk-1, yk-2, ..., y1, y0), yi ≤ xi • P’YES = (yk, xk – yk, xk-1, ..., x2, x1 + y0) • P’NO = (xk, yk, xk-1, ..., x2, x1 + x0 – y0) • End: Paul wins if final position is • P = (0, ..., 0) [Carole has cheated] • P = (0, ..., 0, 1, 0, ..., 0) [Just one possible secret left] Benjamin Doerr Liar Games with Lies in an Interval

  15. Weight Functions • Weight of position P with r rounds remaining: wr(P) = (r – k + 2) 2k-1 xk + 2k-1 xk-1 + 2k-2 xk-2 + ... + x0 • Start: P = (n, 0, ..., 0) has weight wq(P) = (q – k + 2)2k-1n • Each round: wr(P) = wr-1(P’YES) + wr-1(P’NO) → Carole can keep at least half of the weight! • Endgame (r ≤ k): Carole wins iff wk(P) > 2k. Carole wins if wq(PSTART) > 2q. [Our lower bound for ‘n large’] Benjamin Doerr Liar Games with Lies in an Interval

  16. f g ( ) l l l k l k O 2 1 + + + § m a x o g n o g o g n o g n ; Summary and Open Problems • New game: Lies in an interval of k rounds. • Number of rounds necessary to guess the secret • For large n, this is k more than in the one-lie game. • Further work • More precise bounds • More intervals of lies • Other restrictions for the liar [other errors in the communication model] → Spencer’s recent work on half-lies. Thanks! Benjamin Doerr Liar Games with Lies in an Interval

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