2026/08/02 by Matthew Fried
Computer Science · #cs.GT
16 pages
arxiv created 2026/08/02 · arxiv updated 2026/08/04
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 (ℂm)⊗ n. Consequences: the joint information of all anonymous values is a component of dimension polynomial in n against an mn-dimensional game space, with the classical binary theory as the m=2 shadow. Three players can hide: for m≥3 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≥4 the blind space is spanned by ±1 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≥ 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 mn-1 nonzero queries. All dimension and rank claims are verified computationally.