“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)

ADFOCS Benjamin Doerr Liar Games with Lies in an Interval August 21 - August 25, 2006, Saarbrücken, Germany Advanced Course on the Foundations of Computer Science Tamal Dey Joel SpencerIngo 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!

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

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

Liar Games & Noisy Communication Benjamin Doerr Liar Games with Lies in an Interval Satellite Base station Task: Satellite sends data to base station Problem: Transmission errors [noisy communication]. 00 Solution: Allow two-way communication [reciever may confirm/ask particular data]. Assume: No errors on back-wards channel. 0

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). Benjamin Doerr Liar Games with Lies in an Interval Perfect information game: Clear who wins (in theory)

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 n · 2 q = ¡ q · k ¢ n · 2 q =( q + 1 ) n · 2 q

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

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

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] Benjamin Doerr Liar Games with Lies in an Interval Critical value k · l og l ogn q ¼ l ogn + kl og l ognq ¼ l ogn + l og l ogn + k

Let’s Play! (n = 10 ‘secrets’, k = 2 lies) Benjamin Doerr Liar Games with Lies in an Interval 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)

Let’s Play! (n = 10 ‘secrets’, k = 2 lies) Benjamin Doerr Liar Games with Lies in an Interval 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,..., 7: Possible, if lied 1+ rounds ago (no further lie possible) 8, 9, 10: Not possible (lied in round 1 and 3) Position P = (x k,..., x 0 ): x k = # of possible secrets with no lie x i = # of secrets with first lie k-i rounds ago x 0 = # of secrets with no lies allowed P = (10, 0, 0) P = (5, 5, 0) P = (2, 3, 5) P = (1, 1, 5)

Let’s Play! (n = 10 ‘secrets’, k = 2 lies) Benjamin Doerr Liar Games with Lies in an Interval 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,..., 7: Possible, if lied 1+ rounds ago (no further lie possible) 8, 9, 10: Not possible (lied in round 1 and 3) Position P = (x k,..., x 0 ): x k = # of possible secrets with no lie x i = # of secrets with first lie k-i rounds ago x 0 = # of secrets with no lies allowed P = (10, 0, 0) P = (5, 5, 0) P = (2, 3, 5) P = (1, 1, 5) Question Q = (x k,..., x 0 ) Q = (5,0,0) Q = (2,3,0) Q = (1,1,2)

Rules in Vector Format Start: P = (n, 0,..., 0). q Rounds: P = (x k, x k-1, x k-2,..., x 1, x 0 ) Q = (y k, y k-1, y k-2,..., y 1, y 0 ), y i ≤ x i P’ YES = (y k, x k – y k, x k-1,..., x 2, x 1 + y 0 ) P’ NO = (x k, y k, x k-1,..., x 2, x 1 + x 0 – y 0 ) 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

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

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. Benjamin Doerr Liar Games with Lies in an Interval max f l ogn + l og l ogn + k ; l ogn + 2 k g § O ( 1 ) Thanks!