What Multichoice Values Cannot See: The Information Content of Anonymous Values for Games with Graded Participation

Abstract

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.