The Inverse Shapley Value Problem
A. De and I. Diakonikolas and R. Servedio.
39th International Colloquium on Automata, Languages and Programming (ICALP), 2012, pp. 266-277.


For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ Shapley-Shubik Indices of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice theory as a measure of the ``influence'' of voters. The Inverse Shapley Value Problem is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work.

We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant $\eps > 0$ our algorithm runs in fixed poly$(n)$ time (the degree of the polynomial is independent of $\eps$) and has the following performance guarantee: given as input a vector of desired Shapley indices, if any ``reasonable'' weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley indices to within error $\eps.$ If there is a ``reasonable'' voting scheme in which all voting weights are integers at most $\poly(n)$ that approximately achieves the desired Shapley indices, then our algorithm runs in time $\poly(n)$ and outputs a weighted voting scheme that achieves the target vector of Shapley indices to within error $\eps=n^{-1/8}.$

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