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

“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 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!

Overview

Introduction to Liar Games Basic Problem

Motivation: Noisy Communication

History

New Game: Lies in an interval Problem

Result

Example

Some proof details


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.


Liar Games & Noisy Communication


Satellite

Base station

Task: Satellite sends datato base station.

0000

1

0

1

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).


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 · 2q=¡

q· k

¢n · 2q=(q+ 1)

n · 2q

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.


Lies in an Interval: Results

Paul wins if

Carole wins if

Two Cases (left inequalities)

(right inequalities)


q¸ dlogne+ 2kand

or

q¸ dlogne+ k +dloglog2ne

q< logn +2kq< logn +k +loglog2n ¡ 1

k · loglogn

k ¸ loglogn

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

k · loglogn

q¼logn +kloglogn

q¼logn + loglogn +k

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)









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

P = (10, 0, 0)

P = (5, 5, 0)

P = (2, 3, 5)

P = (1, 1, 5)









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

P = (10, 0, 0)

P = (5, 5, 0)

P = (2, 3, 5)

P = (1, 1, 5)

Question Q = (xk, ..., x0)

Q = (5,0,0)

Q = (2,3,0)

Q = (1,1,2)

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

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’]

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

maxf logn + loglogn +k;logn +2kg§ O(1)

Thanks!

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

Documents

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