谎言数固定的Renyi-Ulam病态说谎者博弈
The Renyi-Ulam pathological liar game with a fixed number of lies
摘要：
$q$轮的Renyi-Ulam病态说谎者对策是一个2人完全信息零和对策，其中$k$在集合$[n]:=\{1,...,n\}$上.在每一轮中，Paul选择一个子集$A\subseteq [n]$，Carole为$A$的每个元素或$[n]\setminus A$的每个元素分配1个lie。如果在$q$轮之后，至少有一个元素具有$q$或更少的谎言，则保罗获胜。该游戏是双重的原始Renyi-Ulam说谎者游戏，其获胜条件是最多一个元素具有$k$或更少的谎言。我们证明了保罗的一个获胜策略的存在性的覆盖的离散超立方体与某些放松汉明球的存在性。定义$F^*_k(q)$为最小$n$，使得Paul可以用$k$谎言和初始集合$[n]$赢得$q$轮病态说谎者游戏，我们精确地找到了$F^*_1(q)$和$F^*_2(q)$.对于固定的$k$，我们证明了$F_k^*(q)$是在一个绝对常数（仅取决于$k$）的球界，$2^q/\binom{q}{\leq k}$;这是已知的，以保持原来的雷尼乌拉姆说谎者游戏由于J.斯宾塞的结果。
英文摘要：
The $q$-round Renyi-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 $A\subseteq [n]$ and Carole either assigns 1 lie to each element of $A$ or to each element of $[n]\setminus A$. Paul wins if after $q$ rounds there is at least one element with $k$ or fewer lies. The game is dual to the original Renyi-Ulam liar game for which the winning condition is that at most one element has $k$ or fewer lies. We prove the existence of a winning strategy for Paul to the existence of a covering of the discrete hypercube with certain relaxed Hamming balls. Defining $F^*_k(q)$ to be the minimum $n$ such that Paul can win the $q$-round pathological liar game with $k$ lies and initial set $[n]$, we find $F^*_1(q)$ and $F^*_2(q)$ exactly. For fixed $k$ we prove that $F_k^*(q)$ is within an absolute constant (depending only on $k$) of the sphere bound, $2^q/\binom{q}{\leq k}$; this is already known to hold for the original Renyi-Ulam liar game due to a result of J. Spencer.
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