多选择值无法看到的内容：分级参与博弈的匿名值信息含量
What Multichoice Values Cannot See: The Information Content of Anonymous Values for Games with Graded Participation
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中文总结 AI 辅助
该研究分析分级参与合作博弈的匿名值信息边界，揭示其仅能感知Specht特定分量，存在不可见手性结构，给出交互探测、审计规避的条件及博弈可见部分的高效概要计算方法。
中文摘要 AI 辅助
在具有分级参与的合作博弈中，n个玩家中的每个都在m个有序水平之一上行动；多选择博弈、弃权投票和分级特征归因都属于这种形式。我们针对所有m，精确确定为该场景提出的所有线性、玩家对称值、权力指数和重要性测度（无论当前还是未来）的整个族能看到和不能看到的内容。机制很简短：匿名值下计算任意玩家收益的函数在其他n-1个玩家的排列下不变，根据分支规则，此类不变量仅存在于(ℂ^m)⊗ⁿ的Specht分量(n)和(n-1,1)中。结论：所有匿名值的联合信息是维度随n多项式增长的分量，对应mⁿ维博弈空间，经典二元理论是m=2时的投影。三个玩家可隐藏：当m≥3时，盲空间在n=3时已非零，第一个不可见分量不是相关性而是手性，由三个选民的两种不同单调弃权投票规则实现，每个匿名权力指数对其评分相同。对于n≥4，盲空间由每个含四个轮廓的±1博弈张成。d阶交互探测恰好能看到第一行外至多d个单元的划分，仅当d=n−⌈n/m⌉时可完全恢复。审计规避比二元理论更容易：c个玩家组成的联盟规避所有d阶审计当且仅当c−⌈c/m⌉≥d+1，因此在三个或更多水平下，三人组可对每个基于值的支付方案进行不可见的重构。算法上，博弈的可见部分是多项式大小、易于估计的概要，而任何满支撑值的精确计算需要所有mⁿ−1个非零查询。所有维度和秩的主张均通过计算验证。
英文摘要
In a cooperative game with graded participation, each of $n$ players acts at one of $m$ ordered levels; multichoice games, voting with abstention, and graded feature attribution all take this form. We determine exactly what the entire family of linear, player-symmetric values, power indices, and importance measures proposed for this setting, present and future, can and cannot see, for all $m$ at once. The mechanism is short: the functional computing any one player's payoff under an anonymous value is invariant under permutations of the other $n-1$ players, and by the branching rule such invariants exist only in the Specht constituents $(n)$ and $(n-1,1)$ of $(\mathbb{C}^m)^{\otimes n}$. Consequences: the joint information of all anonymous values is a component of dimension polynomial in $n$ against an $m^n$-dimensional game space, with the classical binary theory as the $m=2$ shadow. Three players can hide: for $m\ge3$ the blind space is nonzero already at $n=3$, and the first invisible constituent is not a correlation but a chirality, realized by two distinct monotone abstention-voting rules on three voters that every anonymous power index scores identically. For $n\ge4$ the blind space is spanned by $\pm1$ games on four profiles each. Order-$d$ interaction probes see exactly the partitions with at most $d$ cells outside the first row, with full recovery only at $d=n-\lceil n/m\rceil$. Audit evasion gets easier than in the binary theory: a coalition of $c$ players evades every order-$d$ audit iff $c-\lceil c/m\rceil\ge d+1$, so with three or more levels, trios can restructure invisibly to every value-based payment scheme. Algorithmically, the visible part of a game is a polynomial-size, cheaply estimable sketch, while exact computation of any full-support value requires all $m^n-1$ nonzero queries. All dimension and rank claims are verified computationally.
