Liar Game Research Articles

Promoting Honesty in Children, or Fostering Pathological Behaviour?

Promoting Honesty in Children, or Fostering Pathological Behaviour?

Formula omitted]-ary Rényi–Ulam pathological liar game with one lie

The m -round q - ary Rényi–Ulam pathological liar game with e lies, referred to as the game [ q , e , n ; m ] ∗ , is considered. Two players, say Paul and Carole, fix nonnegative integers m , n , q and e . In each round, Paul splits [ n ] ≔ { 1 , 2 , … , n } into q subsets, and Carole chooses one subset as her answer and assigns 1 lie to all elements except those in her answer. Paul wins, after m rounds, if there exists at least one element assigned with e or fewer lies. Let f ∗ ( q , e , n ) be the maximum value of m such that Paul can certainly win the game [ q , e , n ; m ] ∗ . This paper gives the exact value of f ∗ ( q , 1 , n ) for n ≥ q q − 1 and presents a tight bound on f ∗ ( q , 1 , n ) for n < q q − 1 .

Linearly bounded liars, adaptive covering codes, and deterministic random walks

We analyze a deterministic form of the random walk on the integer line called the liar machine, similar to the rotor-router model, finding asymptotically tight pointwise and interval discrepancy bounds versus random walk. This provides an improvement in the best-known winning strategies in the binary symmetric pathological liar game with a linear fraction of responses allowed to be lies. Equivalently, this proves the existence of adaptive binary block covering codes with block length n, covering radius ≤ fn for f ∈ (0, 1/2), and cardinality O( √ log log n/(1−2f)) times the sphere bound 2/ ( n ≤bfnc ) .

Two-batch liar games on a general bounded channel

We consider an extension of the 2-person Rényi–Ulam liar game in which lies are governed by a channel C, a set of allowable lie strings of maximum length k. Carole selects x ∈ [ n ] , and Paul makes t-ary queries to uniquely determine x. In each of q rounds, Paul weakly partitions [ n ] = A 0 ∪ ⋯ ∪ A t − 1 and asks for a such that x ∈ A a . Carole responds with some b, and if a ≠ b , then x accumulates a lie ( a , b ) . Carole's string of lies for x must be in the channel C. Paul wins if he determines x within q rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space [ n ] for which Paul can guarantee finding the distinguished element is ∼ t q + k / ( E k ( C ) ( q k ) ) as q → ∞ , where E k ( C ) is the number of lie strings in C of maximum length k. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel C.

A simple solution to Ulam's liar game with one lie

We consider the following game between a questioner and a responder, first proposed by Ulam [9]. (A variation of this game was independently proposed by Renyi, see [5].) Both Renyi and Ulam were motivated by questions arising from communication over noisy channels. The responder thinks of an integer x ∈ {1, . . . , n} and the questioner must determine x by asking questions whose answer is ‘Yes’ or ‘No’. The responder is allowed to lie at most k times during the game. Let qk(n) be the maximum number of questions needed by the questioner, under an optimal strategy, to determine x under these rules. In particular, Ulam asked for the value of q1(10) (as this is related to the well-known ‘twenty questions’ game). It follows from an observation of Berlekamp [1] that q1(10) ≥ 25 and

How to play the one-lie Rényi–Ulam game

The one-lie Rényi–Ulam liar game is a two-player perfect information zero-sum game, lasting q rounds, on the set [ n ] ≔ { 1 , … , n } . In each round Paul chooses a subset A ⊆ [ n ] and Carole either assigns one lie to each element of A or to each element of [ n ] ∖ A . Paul wins the original (resp. pathological) game if after q rounds there is at most one (resp. at least one) element with one or fewer lies. We exhibit a simple, unified, optimal strategy for Paul to follow in both games, and use this to determine which player can win for all q , n and for both games.

The Liar Game Over an Arbitrary Channel

We introduce and analyze a liar game in which t-ary questions are asked and the responder may lie at most k times. As an additional constraint, there is an arbitrary but prescribed list (the channel) of permissible types of lies. For any fixed t, k, and channel, we determine the exact asymptotics of the solution when the number of queries goes to infinity.

The Rényi–Ulam pathological liar game with a fixed number of lies

The q-round Rényi–Ulam pathological liar game with k lies on the set [ n ] ≔ { 1 , … , n } is a 2-player perfect information zero sum game. In each round Paul chooses a subset A ⊆ [ n ] and Carole either assigns 1 lie to each element of A or to each element of [ n ] ⧹ A . Paul wins if after q rounds there is at least one element with k or fewer lies. The game is dual to the original Rényi–Ulam liar game for which the winning condition is that at most one element has k or fewer lies. Define F k * ( q ) to be the minimum n such that Paul can win the q-round pathological liar game with k lies and initial set [ n ] . For fixed k we prove that F k * ( q ) is within an absolute constant (depending only on k) of the sphere bound, 2 q / q ⩽ k ; this is already known to hold for the original Rényi–Ulam liar game due to a result of J. Spencer.

The Two-Batch Liar Game over an Arbitrary Channel

We consider liar games in which player Paul must ask one full batch of questions, receive all answers, and then ask a second and final batch of questions. We show that the effect of this restriction is asymptotically negligible. The strategy for Paul is given explicitly.

Ulam′s pathological liar game with one half‐lie

We introduce a dual game to Ulam′s liar game and consider the case of one half‐lie. In the original Ulam′s game, Paul attempts to isolate a distinguished element by disqualifying all but one of n possibilities with q yes‐no questions, while the responder Carole is allowed to lie a fixed number k of times. In the dual game, Paul attempts to prevent disqualification of a distinguished element by “pathological” liar Carole for as long as possible, given that a possibility associated with k + 1 lies is disqualified. We consider the half‐lie variant in which Carole may only lie when the true answer is “no.” We prove the equivalence of the dual game to the problem of covering the discrete hypercube with certain asymmetric sets. We define for the case k = 1 to be the minimum number n such that Paul can prevent Carole from disqualifying all n elements in q rounds of questions, and prove that .