We study a guessing game where Alice holds a discrete random variable $X$ , and Bob tries to sequentially guess its value. Before the game begins, Bob can obtain side-information about $X$ by asking an oracle, Carole, any binary question of his choosing. Carole’s answer is however unreliable, and is incorrect with probability $\epsilon $ . We show that Bob should always ask Carole whether the index of $X$ is odd or even with respect to a descending order of probabilities – this question simultaneously minimizes all the guessing moments for any value of $\epsilon $ . In particular, this result settles a conjecture of Burin and Shayevitz. We further consider a more general setup where Bob can ask a multiple-choice $M$ -ary question, and then observe Carole’s answer through a noisy channel. When the channel is completely symmetric, i.e., when Carole decides whether to lie regardless of Bob’s question and has no preference when she lies, a similar question about the ordered index of $X$ (modulo $M$ ) is optimal. Interestingly however, the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels. We provide supporting evidence for this difficulty, by showing that a core property required in our proofs becomes NP-hard to test in the general $M$ -ary case. We establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution, which we prove to be NP-hard for $M\geq 3$ .

中文翻译：

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使用不可靠的 Oracle 进行最少的猜测

我们研究了一个猜谜游戏，其中 Alice 持有一个离散随机变量 $X$ ，Bob 尝试依次猜测它的值。在游戏开始之前，Bob 可以获得关于 $X$ 通过询问预言机 Carole，他选择的任何二元问题。然而，卡罗尔的回答不可靠，而且概率不正确 $\epsilon $ . 我们证明 Bob 应该总是问 Carole $X$ 就概率的降序而言是奇数或偶数——这个问题同时最小化了任何值的所有猜测时刻 $\epsilon $ . 尤其是，这个结果解决了 Burin 和 Shayevitz 的一个猜想。我们进一步考虑了一个更通用的设置，Bob 可以在其中进行多项选择 百万美元 -ary 问题，然后通过嘈杂的频道观察 Carole 的回答。当信道完全对称时，即当 Carole 不考虑 Bob 的问题而决定是否撒谎，并且在撒谎时没有偏好时，类似的问题是关于 $X$ （模数 百万美元 ) 是最优的。然而有趣的是，在其他对称渠道中，测试给定问题是否最优的问题似乎通常很困难。我们为这个困难提供了支持证据，通过证明我们证明中所需的核心属性在一般情况下变得难以测试 百万美元 -ary 案例。我们通过减少测试模差分失等系统是否有解的问题来建立这个硬度结果，我们证明它是 NP 难的 $M\geq 3$ .