Reconstructing weighted voting schemes from partial information about their power indices

Emmanouil-Vasileios Vlatakis-Gkaragkounis, Anindya De, Rocco A Servedio, Huck Bennett

Abstract:

A number of recent works [Gol06, OS11, DDS17, DDFS14] have considered the problem
of approximately reconstructing an unknown weighted voting scheme given information about
various sorts of “power indices” that characterize the level of control that individual voters have
over the final outcome. In the language of theoretical computer science, this is the problem of
approximating an unknown linear threshold function (LTF) over {−1, 1}
n given some numerical
measure (such as the function’s n “Chow parameters,” a.k.a. its degree-1 Fourier coefficients, or
the vector of its n Shapley indices) of how much each of the n individual input variables affects
the outcome of the function.
In this paper we consider the problem of reconstructing an LTF given only partial information
about its Chow parameters or Shapley indices; i.e. we are given only the Chow parameters or
the Shapley indices corresponding to a subset S ⊆ [n] of the n input variables. A natural goal
in this partial information setting is to find an LTF whose Chow parameters or Shapley indices
corresponding to indices in S accurately match the given Chow parameters or Shapley indices
of the unknown LTF. We refer to this as the Partial Inverse Power Index Problem.
Our main results are a polynomial time algorithm for the (ε-approximate) Chow Parameters Partial Inverse Power Index Problem and a quasi-polynomial time algorithm for the (εapproximate) Shapley Indices Partial Inverse Power Index Problem.