On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games

TL;DR

This paper investigates the computational difficulty of designing weighted voting games to match specified influence levels, proving that the inverse problem is generally intractable for many semivalues including Banzhaf and Shapley indices.

Contribution

It establishes the intractability of the inverse semivalue problem for a broad class of power indices, including the most popular ones.

Findings

Inverse problem is computationally intractable for many semivalues.

Hardness results include Banzhaf and Shapley indices.

Results highlight fundamental computational limits in voting game design.

Abstract

Weighted voting games are a family of cooperative games, typically used to model voting situations where a number of agents (players) vote against or for a proposal. In such games, a proposal is accepted if an appropriately weighted sum of the votes exceeds a prespecified threshold. As the influence of a player over the voting outcome is not in general proportional to her assigned weight, various power indices have been proposed to measure each player's influence. The inverse power index problem is the problem of designing a weighted voting game that achieves a set of target influences according to a predefined power index. In this work, we study the computational complexity of the inverse problem when the power index belongs to the class of semivalues. We prove that the inverse problem is computationally intractable for a broad family of semivalues, including all regular semivalues. As a special case of our general result, we establish computational hardness of the inverse problem for the Banzhaf indices and the Shapley values, arguably the most popular power indices.

Equations

$${\tilde{f}}^{p^n}(i)=\sum _{x\in \{-1,1{\}}^n:x_i=-1}{p}_{wt(x)}^{n}f(x)x_i+\sum _{x\in \{-1,1{\}}^n:x_i=1}{p}_{wt(x)-1}^{n}f(x)x_i,$$

$${\tilde{f}}^{p^n}(i)$$

$${\mu }_{p^n}(x):=\frac{{\mu }_{p^n}^{\prime }(x)}{\Lambda (p^n)},$$

$${\mathcal{A}}_{p^{n+2}}:=\left\{c\in {\mathbb{R}}^{n+2}:\exists w\in {\mathbb{R}}^{n+2},w_i>0,1\le i\le n,w_{n+1}=w_{n+2}=-\sum _{i=1}^n{w}_i/2,c_i={\tilde{f}}_{w,0}^{p^{n+2}}(i),1\le i\le n+2\right\}.$$

$${\text{Pr}}_{x\sim {\mu }_{p^{n+2}}}[w\cdot x=0]$$

$$+\frac{|\{x:w\cdot x=0,x\notin \{\text{-1},\text{1}\}\}|}{2\Lambda (p^{n+2})}(p_{k+1}^{n+2}+p_k^{n+2})+\frac{1}{\Lambda (p^{n+2})}(p_{n+1}^{n+2}+p_0^{n+2}),$$

$$\left|\left\{S\subset \{1,\dots ,n\}:\sum _{i\in S}{c}_i=\sum _{i=1}^n{c}_i/2\right\}\right|=\frac{\Lambda (p^{n+2}){\text{Pr}}_{x\sim {\mu }_{p^{n+2}}}[w\cdot x=0]-(p_{n+1}^{n+2}+p_0^{n+2})}{(p_{n-k+1}^{n+2}+p_{n-k}^{n+2}+p_{k+1}^{n+2}+p_k^{n+2})}.$$

$$c$$

$$d$$

$$e$$

$$|d\cdot x|=\{\begin{array}{ll}|{\sum }_{i=1}^n{a}_i{x}_i+A-2y|,& \text{ if }{x}_1=1,x_{n+1}=x_{n+2}=-1\\ |{\sum }_{i=1}^n{a}_i{x}_i+A|,& \text{ if }{x}_1=-1,x_{n+1}=x_{n+2}=-1\\ |{\sum }_{i=1}^n{a}_i{x}_i-A|,& \text{ if }{x}_1=1,x_{n+1}=x_{n+2}=1\\ |{\sum }_{i=1}^n{a}_i{x}_i-A+2y|,& \text{ if }{x}_1=-1,x_{n+1}=x_{n+2}=1\\ |{\sum }_{i=1}^n{a}_i{x}_i-y|,& \text{ if }{x}_1=1,x_{n+1}\ne x_{n+2}\\ |{\sum }_{i=1}^n{a}_i{x}_i+y|,& \text{ if }{x}_1=-1,x_{n+1}\ne x_{n+2}\end{array}$$

$$|e\cdot x|=\{\begin{array}{ll}|{\sum }_{i=1}^n{a}_i{x}_i+A+2y|,& \text{ if }{x}_1=1,x_{n+1}=x_{n+2}=-1\\ |{\sum }_{i=1}^n{a}_i{x}_i+A|,& \text{ if }{x}_1=-1,x_{n+1}=x_{n+2}=-1\\ |{\sum }_{i=1}^n{a}_i{x}_i-A|,& \text{ if }{x}_1=1,x_{n+1}=x_{n+2}=1\\ |{\sum }_{i=1}^n{a}_i{x}_i-A-2y|,& \text{ if }{x}_1=-1,x_{n+1}=x_{n+2}=1\\ |{\sum }_{i=1}^n{a}_i{x}_i+y|,& \text{ if }{x}_1=1,x_{n+1}\ne x_{n+2}\\ |{\sum }_{i=1}^n{a}_i{x}_i-y|,& \text{ if }{x}_1=-1,x_{n+1}\ne x_{n+2}\end{array}$$

$$|c\cdot x|=|e\cdot x|=|d\cdot x|=0.$$

$$|d\cdot x|+|e\cdot x|=|c\cdot x|.$$

$$|d\cdot x|+|e\cdot x|=|c\cdot x|,$$

$$|d\cdot x|+|e\cdot x|=2y.$$

$$|d\cdot x|+|e\cdot x|=|c\cdot x|,$$

$$K_{{\mu }_{p^{n+2}}}(d)+K_{{\mu }_{p^{n+2}}}(e)-K_{{\mu }_{p^{n+2}}}(c)={\mathbb{E}}_{x\sim {\mu }_{p^{n+2}}}\left[|d\cdot x|\right]+{\mathbb{E}}_{x\sim {\mu }_{p^{n+2}}}\left[|e\cdot x|\right]-{\mathbb{E}}_{x\sim {\mu }_{p^{n+2}}}\left[|c\cdot x|\right]=\sum _{x\in \{-1,1{\}}^{n+2}}2y\cdot {\mu }_{p^{n+2}}(x)\cdot \text{mathds}{1}_{c\cdot x=0,x\notin \{-1,1\}}.$$

$${\mathrm{Pr}}_{x\sim {\mu }_{p^{n+2}}}[a\cdot x=0]=$$

$$=\frac{K_{{\mu }_{p^{n+2}}}(d)+K_{{\mu }_{p^{n+2}}}(e)-K_{{\mu }_{p^{n+2}}}(c)}{2y}+\frac{p_0+p_{n+1}}{\Lambda (p^{n+2})}.$$

$$a\cdot c$$

$$=\frac{\Lambda (p^{n+2})}{2}\sum _{x\in \{-1,1{\}}^{n+2}}f(x){\mu }_{p^{n+2}}(x)\sum _{i=1}^{n+2}{a}_i{x}_i$$

$$\le \frac{\Lambda (p^{n+2})}{2}\sum _{x\in \{-1,1{\}}^{n+2}}{\mu }_{p^{n+2}}(x)\left|\sum _{i=1}^{n+2}{a}_i{x}_i\right|,$$

$$a\cdot c=\frac{\Lambda (p^{n+2})}{2}\sum _{x\in \{-1,1{\}}^{n+2}}{\mu }_{p^{n+2}}(x)\left|\sum _{i=1}^{n+2}{a}_i{x}_i\right|,$$

$${\mathbb{E}}_{x\sim {\mu }_{p^{n+2}}}\left[\left|\sum _{i=1}^{n+2}{a}_i{x}_i\right|\right]=\frac{2{\max }_{c\in {\mathcal{C}}_{p^{n+2}}}a\cdot c}{\Lambda (p^{n+2})}.$$

$${\tilde{g}}^{p^{n+2}}(i)=\sum _{x:x_i=-1}{p}_{wt(x)}^{n+2}(g(x^i)-g(x))=\sum _{\begin{array}{c}x:x_i=-1,\\ x_{n+1}=x_{n+2}=-1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j+a_i-A\right)-\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j-a_i-A\right)\right)+\sum _{\begin{array}{c}x:x_i=-1,\\ x_{n+1}=x_{n+2}=1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j+a_i+A\right)-\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j-a_i+A\right)\right)$$

$$+\sum _{\begin{array}{c}x:x_i=-1,\\ x_{n+1}\ne x_{n+2}\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j+a_i\right)-\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j-a_i\right)\right)$$

$${\tilde{f}}^{p^{n+2}}(i)=\sum _{x:x_i=-1}{p}_{wt(x)}^{n+2}(f(x^i)-f(x))=\sum _{\begin{array}{c}x:x_i=-1,\\ x_{n+1}=x_{n+2}=1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j+a_i-A\right)-\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j-a_i-A\right)\right)+\sum _{\begin{array}{c}x:x_i=-1,\\ x_{n+1}=x_{n+2}=-1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j+a_i+A\right)-\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j-a_i+A\right)\right)+\sum _{\begin{array}{c}x:x_i=-1,\\ x_{n+1}\ne x_{n+2}\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j+a_i\right)-\mathrm{sign}\left(\sum _{j\le n:j\ne i}{a}_j{x}_j-a_i\right)\right).$$

$${\tilde{f}}^{p^{n+2}}(i)-{\tilde{g}}^{p^{n+2}}(i)=2(p_{n+1}^{n+2}-p_{n-1}^{n+2}).$$

$${\tilde{g}}^{p^{n+2}}(n+1)=\sum _{x:x_{n+1}=-1}{p}_{wt(x)}^{n+2}(g(x^{n+1})-g(x))=\sum _{\begin{array}{c}x:x_{n+1}=-1,\\ x_{n+2}=-1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j\right)-\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j-A\right)\right)+\sum _{\begin{array}{c}x:x_{n+1}=-1,\\ x_{n+2}=1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j+A\right)-\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j\right)\right).$$

$${\tilde{f}}^{p^{n+2}}(n+1)=\sum _{x:x_{n+1}=-1}{p}_{wt(x)}^{n+2}(f(x^{n+1})-f(x))=\sum _{\begin{array}{c}x:x_{n+1}=-1,\\ x_{n+2}=1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j-A\right)-\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j\right)\right)+\sum _{\begin{array}{c}x:x_{n+1}=-1,\\ x_{n+2}=-1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j\right)-\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j+A\right)\right).$$

$${\tilde{f}}^{p^{n+2}}(n+1)-{\tilde{g}}^{p^{n+2}}(n+1)=\sum _{\begin{array}{c}x:x_{n+1}=-1,\\ x_{n+2}=-1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j-A\right)-\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j+A\right)\right)+\sum _{\begin{array}{c}x:x_{n+1}=-1,\\ x_{n+2}=1\end{array}}{p}_{wt(x)}^{n+2}\left(\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j-A\right)-\mathrm{sign}\left(\sum _{j\le n}{a}_j{x}_j+A\right)\right)=-2\sum _{t=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{t}\right)(p_t^{n+2}+p_{t+1}^{n+2}).$$

$${\tilde{f}}^{p^{n+2}}(n+2)-{\tilde{g}}^{p^{n+2}}(n+2)=-2\sum _{t=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{t}\right)(p_t^{n+2}+p_{t+1}^{n+2}).$$

Full text

1 Introduction

1.1 Background and Motivation

Weighted voting games are a classical family of cooperative games that have been extensively studied in the game theory and social choice literature. Such games model a common voting scenario where each agent (player), associated with a weight, casts a “YES” (for) or “NO” (against) vote: if the weighted sum of the “YES” votes exceeds a threshold, then the voting outcome is “YES”, otherwise the outcome is “NO”. Examples of such practical scenarios include the voting system of the European Union, stockholder companies, and resource allocation in multi-agent systems [EGGW08, DKKZ14].

Although having a larger weight might help an agent affect the voting outcome, her influence on the result of the game is not always proportional to her weight. Thus, instead of using agents’ weights, the power of an agent over the outcome is usually measured in a systematic way by a power index. Over the years, many power indices have been proposed and studied, such as the Shapley value [Sha53] (also known as Shapley-Shubik index for weighted voting games [SS54]), the Banzhaf index [BI64], the Deegan-Packel index [DP78], and the Holler index [Hol82]. The problem of computing the agents’ power indices in a given game has received ample attention and its computational complexity is well-understood for many game representations and power index functions (see, e.g.,  [PK90, DP94, Azi08]).

1.2 Our Contributions

In this work, we focus on the inverse power index problem — that is, the problem of designing a weighted voting game with a given set of power indices. As we will explain in detail below, the inverse problem has been extensively studied in various fields, including game theory, social choice theory, and learning theory. Various works have provided heuristic methods, exponential time algorithms, or polynomial time approximation algorithms with provable performance guarantees for this problem.

Despite this wealth of prior work on the algorithmic version of the inverse problem, prior to this work its computational complexity was not well-understood, even for the most popular power indices (Shapley values, Banzhaf indices). In this paper, we study and essentially resolve the computational complexity of the inverse power index problem for weighted voting games, with respect to a broad and extensively studied family of power indices. Specifically, we show that the inverse problem is computationally intractable under standard complexity assumptions. More specifically, we prove that for a large class of power indices — that includes the popular Banzhaf index, Shapley values, and the class of semivalues [Web79] — the inverse problem cannot be in the polynomial hierarchy (PH), unless the polynomial hierarchy collapses. Prior to this work, it was conceivable that there exists an exact polynomial time algorithm for this problem. It follows from our hardness result that the existence of such an algorithm is unlikely.

1.3 Related work

Several heuristic algorithms for the inverse Banzhaf index problem have been proposed in the social choice and game theory literature. Aziz, Paterson, and Leech [APL07] give an approximation algorithm that on input target Banzhaf indices and a desired $\ell_{2}$-distance bound outputs a weighted voting game with integer weights that has Banzhaf indices within the desired distance bound. Unfortunately, no theoretical guarantees are provided regarding the convergence rate of this method, and it is not known whether it converges to an approximately optimal solution. Two related heuristic algorithms that return weighted voting games with Banzhaf indices within a given distance from the target indices are proposed by Laruelle and Widgren [LW98] and Leech [Lee03]. Similarly to [APL07], Fatima, Wooldridge, and Jennings [FWJ08] give an iterative approximation algorithm for the inverse Shapley value problem that on input target Shapley values, a quota, and a desired average percentage difference, outputs a weighted voting game with the given quota that has Shapley values within the desired distance. It is shown that each iteration runs in quadratic time and that the algorithm eventually converges, but no theoretical guarantees are given regarding the convergence rate, i.e., the time until a desired approximation is achieved.

An exact algorithm for both the inverse Banzhaf index and inverse Shapley value problems is given by Kurz [Kur12]. The proposed method relies on integer linear programming and returns a weighted voting game that minimizes the $\ell_{1}$-distance from the target power indices. This method has running time exponential in the number of players. Another exact, but exponential-time, algorithm for the inverse power index problem is given by De Keijzer, Klos, and Zhang [DKKZ14]. The [DKKZ14] algorithm outputs a weighted voting game in which the power indices of the players are as close to the target vector as possible. The presented algorithm is based on an enumeration of all weighted voting games: for each weighted voting game, the algorithm computes the power indices, their distance from the target ones and it then outputs the game with the smallest distance. Since there exist $2^{\Omega(n^{2})}$ weighted voting games with $n$ players, this algorithm also runs in exponential time.

In addition to the several heuristics and exponential time algorithms that have been proposed, a line of recent works in theoretical computer science [OS11, DDFS14, DDS17] have obtained polynomial time approximation algorithms with provable performance guarantees for the inverse problem with respect to both the Banzhaf indices and the Shapley values. These algorithms output a weighted voting game whose power indices have small $\ell_{2}$-distance from the target indices. These algorithmic results were recently extended to more general classes of functions [DK18].

The inverse power index problem is also of significant interest in various other fields, such as circuit complexity and computational learning theory. The reader is referred to [OS11] for a detailed summary of this connection. In the fields of computational complexity and learning theory, linear threshold functions (LTFs), which are equivalent to weighted voting games if we allow negative weights [DKKZ14], have been of great significance and have been studied for several decades [Ros58, Cho61, MTT61b, Der65, MP68]. A fundamental result of C. K. Chow from the the early 1960s [Cho61] shows that linear threshold functions are characterized by their degree-[math] and degree-$1$ “Fourier coefficients”, now known as Chow parameters. Given this structural result, the following natural computational question — now known as “the Chow parameters problem” [OS11] — arises: Given the Chow parameters of an LTF, reconstruct a weights-based representation of the function. Interestingly, the Chow parameters are essentially equivalent to the non-normalized Banzhaf indices [DS79], and therefore the inverse Banzhaf index problem is tantamount to the Chow parameters problem.

In addition to the aforementioned algorithmic results, a number of complexity results have been established concerning weighted voting games. Aziz [Azi08], Elkind et al. [EGGW09], and Elkind et al. [EGGW08] study the computational complexity of various problems related to weighted voting games. Aziz [Azi08] studies the complexity of computing various indices such as the Shapley values, the Banzhaf, and the Deegan-Packel indices for a given simple game when the game is given in different forms. Finally, Faliszewski and Hemaspaandra [FH08] study the complexity of the power index comparison problem: given two weighted voting games and a player, decide on which game the given player has higher influence as it is computed by a specific power index. They show that this problem is intractable, namely PP-complete, for both the Shapley values and Banzhaf index. To achieve this, they extend the #P-metric-completeness of computing the Shapley values, proved by Deng and Papadimitriou [DP94]. They prove that, whereas computing the Banzhaf indices of a weighted voting game is #P-parsimonious-complete [PK90], computing the Shapley values is #P-many-one complete and it cannot be strengthened to #P-parsimonious-complete. Gopalan, Nisan, and Roughgarden [GNR15] study the convex polytope consisting of the Chow parameters of all Boolean functions. They show that the linear optimization problem over this polytope is #P-hard; a result that indicates, but does not logically imply, that the inverse Banzhaf index problem may be intractable.

2 Preliminaries

Notation

We write $wt(x)$ to denote the weight of a Boolean vector $x\in\{-1,1\}^{n}$, i.e., the number of 1’s in $x$. For any $i\in\{1,\dots,n\}$ and $x\in\{-1,1\}^{n}$ such that $x_{i}=-1$, we write $x^{i}$ to denote the vector obtained when we flip the $i^{th}$ coordinate of $x$. We denote by 1 (resp. -1) the vector in $\{-1,1\}^{n}$ with all coordinates equal to $1$ (resp. $-1$). We will denote by $\mathrm{Conv}(S)$ the convex hull of the set $S$. We will use $\mathrm{sign}:\mathbb{R}\rightarrow\mathbb{R}$ for the function that takes value $1$ if $z\geq 0$ and value $-1$ if $z<0$.

Our basic object of study is the family of linear threshold functions (LTFs) over $\{-1,1\}^{n}$:

Definition 1** (Linear Threshold Function).**

A linear threshold function (LTF) is any function $f_{w,\theta}:\{-1,1\}^{n}\rightarrow\{-1,1\}$ such that $f_{w,\theta}(x)=\mathrm{sign}(w\cdot x-\theta)$ for some weight vector $w\in\mathbb{R}^{n}$ and threshold $\theta\in\mathbb{R}$.

Note that weighted voting games are equivalent to LTFs with non-negative weights. We leverage this equivalence throughout this paper. At various points, we may refer to a weighted voting game as an LTF without further elaboration.

Semivalues

We mainly focus on power indices that belong to the class of semivalues. Semivalues are a fundamental family of power indices, introduced by Weber [Web79] as generalizations of the Shapley value that do not satisfy the efficiency axiom [DNW81]. Since their introduction, semivalues have received considerable attention, see, e.g., [Ein87, CFP03, CF08].

We start by providing the definition of semivalues [SR88] in terms of weighting coefficients, as they were characterized by Dubey, Neyman, and Weber [DNW81]:

Definition 2** (Semivalues).**

For a positive integer $n$, a probability vector $p^{n}=(p_{0}^{n},\dots,p_{n-1}^{n})\in\mathbb{R}^{n}$ is a vector such that $\sum_{t=0}^{n-1}\binom{n-1}{t}p_{t}^{n}=1$ and $p_{t}^{n}\geq 0,\text{ for }t\in\{0,\dots,n-1\}$. The $i$-th semivalue corresponding to the probability vector $p^{n}$ of a Boolean function $f:\{-1,1\}^{n}\rightarrow\{-1,1\}$ is defined to be

[TABLE]

for $i\in\{1,\dots,n\}$.

Intuitively, we can interpret $p^{n}$ as the vector of probabilities that a given player will join a coalition of size $t$ [CF08], $0\leq t\leq n-1$. With this interpretation, the $i$-th semivalue computes the probability of the event that player $i$ is a pivot, i.e., the probability that the output of the game would change from $1$ to $-1$ if the $i$-th player (the $i$-th variable) were to change her vote from $1$ to $-1$.

We call a semivalue regular if it is defined by strictly positive probability vectors [CF08].

Remark

The Shapley values and Banzhaf indices are the semivalues defined by $p^{n}_{t}=\frac{(n-t-1)!t!}{n!}$ [SS54] and $p^{n}_{t}=\frac{1}{2^{n-1}}$ [DS79], respectively. Note that both indices are regular semivalues.

Reformulation of Semivalues

The following equivalent way to express a set of semivalues will be useful throughout this paper. Setting $p_{-1}^{n}=p_{n}^{n}=0$, we observe that we can rewrite the semivalues vector defined by the probability vector $p^{n}$ as follows. For $i\in\{1,\dots,n\}$, we have:

[TABLE]

From this representation, a probability distribution $\mu_{p^{n}}$ over $\{-1,1\}^{n}$ emerges, defined as follows. For $x\in\{-1,1\}^{n}$, we have:

[TABLE]

where $\mu^{\prime}_{p^{n}}(x):=p^{n}_{wt(x)}+p^{n}_{wt(x)-1}$ and $\Lambda(p^{n}):=\sum_{x\in\{-1,1\}^{n}}\mu^{\prime}_{p^{n}}(x)$ is the normalizing factor.

For a Boolean function $f:\{-1,1\}^{n}\rightarrow\{-1,1\}$, we will write $\widehat{f}^{p^{n}}(i):=\sum_{x\in\{-1,1\}^{n}}\mu^{\prime}_{p^{n}}(x)f(x)x_{i}$ for the first term of (1) and $C_{f}^{p^{n}}:=\sum_{x\in\{-1,1\}^{n}}f(x)(p^{n}_{wt(x)-1}-p^{n}_{wt(x)})$ for the second term of (1). One can view the first term, $\widehat{f}^{p^{n}}(i)$, as the expectation $\mathbb{E}_{x\sim\mu_{p^{n}}}[f(x)x_{i}]$ (up to the normalizing factor $\Lambda(p^{n})$).

We now define the notion of a reasonable probability vector to describe the family of semivalues for which our computational hardness results apply:

Definition 3** (Reasonable Probability Vector).**

A probability vector $p^{n}\in\mathbb{R}^{n}$ is called reasonable if there exists a $t$ with $t=\Omega(n)$ and $n-t=\Omega(n)$ such that $p^{n}_{t}>0$.

The intuition behind the above definition is that the distribution $\mu_{p^{n}}$ has support $2^{\Omega(n)}$. Note that this happens when there exists a $t$ with $\binom{n}{t}=2^{\Omega(n)}$ such that $p^{n}_{t}>0$. We recall that all regular semivalues (including the Banzhaf indices and Shapley values) satisfy this property.

Remark 1**.**

For computational purposes, throughout this paper, we will assume that each value $p^{n}_{t}$ defining our probability vector is a rational number that can be described as a ratio of integers with $\mathrm{poly}(n)$ bits.**

Inverse Semivalues Problem

We are ready to define the inverse semivalues problem. Let $n$ denote the number of players and consider the semivalues defined by a known probability vector $p^{n}$. Given a vector $c=(c_{1},\ldots,c_{n})$ of target semivalues, we want to either find a weighted voting game with these target semivalues or decide that there does not exist any weighted voting game with semivalues vector $c$.

Name: SV${}_{p^{n}}$-Inverse Problem

Input: A vector $(c_{1},\dots,c_{n})\in\mathbb{Q}^{n}$ and $\theta\in\mathbb{Q}$.

Question: Output $w\in\mathbb{Q}_{+}^{n}$ with $\sum_{i=1}^{n}w_{i}=1$ such that $\widetilde{f}_{w,\theta}^{p^{n}}(i)=c_{i}$, for $i\in\{1,\dots,n\}$, or “NO” if no such $w$ exists.

3 Main Result: Computational Intractability of Inverse Power-Index Problem

3.1 Statement of Main Result and Proof Overview

The main result of this paper is the following:

Theorem 1** (Main Result, Informal Statement).**

For semivalues defined by the probability vector $p^{n}$, if $p^{n}$ is a reasonable probability vector, then the SV${}_{p^{n}}$-Inverse problem is not in the polynomial hierarchy, unless the polynomial hierarchy collapses.

As an immediate corollary of Theorem 1, we obtain that the inverse power index problem is similarly intractable for the class of regular semivalues, which includes the Shapley values and the Banzhaf indices.

Proof Overview

We start with a brief overview of our proof establishing Theorem 1. To prove hardness of the inverse problem, we examine the convex polytope $\mathcal{C}_{p^{n+2}}$ consisting of the convex combinations of the semivalues of linear threshold functions with zero threshold and weight vectors of a specific form described below (Definition 4). We prove (Theorem 2) that if the probability vector $p^{n}$ defining the semivalues is reasonable, then the linear optimization over $\mathcal{C}_{p^{n+2}}$ is #P-hard (under Turing reductions). Then, we proceed to show (Theorem 4) that the optimization problem can be solved using an oracle for the semivalues verification problem, i.e., the problem of verifying that the given target semivalues are the actual semivalues of a given linear threshold function, or an oracle for the inverse problem for weight vectors of the aforementioned specific form. We thus conclude that the verification and the inverse semivalues problem for linear threshold functions with this specific weight structure cannot be in the polynomial hierarchy. Finally, using a lemma that shows that semivalues characterize the space of linear threshold functions with the same threshold (Lemma 3), we show hardness of the inverse and verification problems for linear threshold functions with positive weights, i.e., for weighted voting games, as desired.

Our proof strategy bears some similarities to the approach by Gopalan, Nisan, and Roughgarden [GNR15] (also exploited by Dughmi and Xu [DX16]). In particular, Gopalan, Nisan, and Roughgarden [GNR15] show that the linear optimization problem over the polytope consisting of the Chow parameters of all Boolean functions is #P-complete, and therefore there cannot exist an efficient membership oracle for this polytope. We note that our results include the results of [GNR15] regarding Chow parameters as a very special case: as previously mentioned, the non-normalized Banzhaf indices of a linear threshold function are equal to its Chow parameters and they are semivalues.

Despite this similarity, our proof involves a number of novel ideas that seem necessary in order to handle a broad range of probability distributions that could define a semivalue. One of the difficulties comes from the fact that we want to prove hardness for the class of weighted voting games, i.e., LTFs with positive weights. While this requirement is easy to handle for the Banzhaf indices (Chow parameters), it poses non-trivial difficulties for more general semivalues. To handle this, we propose a generalization of the definition of the Khintchine constant from the uniform distribution to any probability distribution and establish that it is hard to compute under a restricted set of weights that is crucial for our proof (Theorem 3). Another crucial ingredient of our proof is a new structural result (Lemma 3) establishing that the set of semivalues uniquely determines a weighted voting game.

3.2 Proof of Main Result

In this subsection, we proceed with the detailed proof of Theorem 1.

Semivalues Polytope

Our analysis makes essential use of the convex polytope defined as the convex hull of the set of semivalues for all linear threshold functions whose weights-based representation is of a specific form: Namely, their threshold $\theta=0$ and their weight vectors consist of $n$ positive coordinates and two coordinates each of whose weights is equal to minus a half times the sum of the first $n$ coordinates. Formally, we introduce the following definition:

Definition 4**.**

For a positive integer $n$, define $\mathcal{C}_{p^{n+2}}:=\mathrm{Conv}(\mathcal{A}_{p^{n+2}})$, where

[TABLE]

The first main step of our proof involves showing that the linear optimization problem over the above defined polytope is computationally hard.

Linear Optimization over $\mathcal{C}_{p^{n+2}}$

We firstly prove that if the semivalues’ probability distribution defined by $p^{n+2}$ has a sufficiently large support (in particular, if $p^{n+2}$ is a reasonable vector as in Definition 3), then the linear optimization problem over the polytope $\mathcal{C}_{p^{n+2}}$ is #P-hard.

The linear optimization problem for semivalues defined by any probability vector $p^{n+2}$ is captured by the following family of problems:

Name: SV${}_{p^{n+2}}$-Optimization Problem

Input: A vector $a=(a_{1},\dots,a_{n+2})\in\mathbb{Q}^{n+2}$.

Question: Compute $\max_{c\in\mathcal{C}_{p^{n+2}}}a\cdot c$.

The main result of this subsection is the following:

Theorem 2**.**

If $p^{n+2}$ is a reasonable probability vector, the SV${}_{p^{n+2}}$-Optimization Problem is #P-hard.

We prove Theorem 2 by reducing from an intermediate problem — that of computing the Khintchine constant of a vector with respect to the probability distribution $\mu_{p^{n}}$:

Name: Khintchine${}_{\mu_{p^{n}}}$

Input: A vector $a=(a_{1},\dots,a_{n})\in\mathbb{Q}^{n}$.

Question: Compute $K_{\mu_{p^{n}}}(a)=\mathbb{E}_{x\sim\mu_{p^{n}}}[|a\cdot x|]$.

The Khintchine constant has been extensively studied with respect to the uniform distribution on the Boolean hypercube (see, e.g., [Sza76, DDS16] and references therein). We note that [GNR15] established the intractability of computing this quantity under the uniform distribution. We show:

Theorem 3**.**

If $p^{n+2}$ is a reasonable probability vector, the Khintchine${}_{\mu_{p^{n+2}}}$ problem is #P-hard, even restricted to inputs $(a_{1},\dots,a_{n},-A/2,-A/2)$, where $A=\sum_{i=1}^{n}a_{i}$ and $a_{i}>0$ for $i\in\{1,\dots,n\}$.

Proof.

We start by showing that the #Partition problem for the distribution $\mu_{p^{n}}$ is hard and then reduce the latter problem to the former.

Name: #Partition${}_{\mu_{p^{n}}}$

Input: A vector $w=(w_{1},\dots,w_{n})\in\mathbb{Z}^{n}$.

Question: Compute $\text{Pr}_{x\sim\mu_{p^{n}}}[w\cdot x=0]$.

We start with the following proposition:

Proposition 1**.**

If $p^{n+2}$ is a reasonable probability vector, #Partition${}_{\mu_{p^{n+2}}}$ is #P-hard, even restricted to inputs $(w_{1},\dots,w_{n},-W/2,-W/2)$, where $W=\sum_{i=1}^{n}w_{i}$ and $w_{i}>0$ for $1\leq i\leq n$.

Proof.

We reduce from the following problem:

Name: #R-Partition

Input: Positive integers $c_{j},1\leq j\leq n$ and a positive integer $b_{1}n\leq k\leq b_{2}n$, where $0<b_{1}\leq b_{2}<1$, such that if there is a subset $S$ of $\{1,\dots,n\}$ with $\sum_{i\in S}c_{i}=\sum_{i=1}^{n}c_{i}/2$, then $|S|=k$ or $|S|=n-k$.

Question: Compute the number of subsets $S$ of $\{1,\dots,n\}$ such that $\sum_{i\in S}c_{i}=\sum_{i=1}^{n}c_{i}/2$.

Given an instance $c_{1},\dots,c_{n}$ of the #P-complete #R-Partition [DP94], we construct an instance of #Partition${}_{\mu_{p^{n+2}}}$ as follows: We set $w_{i}=c_{i}$, for $1\leq i\leq n$, and define $w=(w_{1},\dots,w_{n},-W/2,-W/2)$, where $W=\sum_{i=1}^{n}w_{i}$. We have:

[TABLE]

as the #R-Partition problem guarantees that every solution has size $k$ or $n-k$ and the number of solutions with size $k$ are equal to the number of solutions with size $n-k$. That is, every $x\in\{-1,1\}^{n+2}\setminus\{\textbf{-1},\textbf{1}\}$ such that $w\cdot x=0$ is guaranteed to have weight $k+1$ or weight $n-k+1$ as $x_{n+1}$ has to be different than $x_{n+2}$, otherwise the only solutions are $\{\textbf{-1},\textbf{1}\}$. So, for every solution of the #R-partition problem we have two $x\in\{-1,1\}^{n+2}\setminus\{\textbf{1},\textbf{-1}\}$ such that $w\cdot x=0$. Thus,

[TABLE]

This completes the proof of Proposition 1. ∎

Given an instance of #Partition${}_{\mu_{p^{n+2}}}$, i.e., $a=(a_{1},\dots,a_{n},-A/2,-A/2)$, where $A=\sum_{i=1}^{n}a_{i}$ and $a_{i}>0$ for $1\leq i\leq n$, we construct the following three Khintchine${}_{\mu_{p^{n+2}}}$ instances:

[TABLE]

where $0<y<1/2<a_{1}$. We will show that solving the above instances of Khintchine${}_{\mu_{p^{n+2}}}$ suffices to solve our given instance of #Partition${}_{\mu_{p^{n+2}}}$. This will complete the proof of Theorem 3. To do so, we require some case analysis and explicit calculations.

For any $x\in\{-1,1\}^{n+2}$, we have that:

[TABLE]

and similarly

[TABLE]

Given the above, we observe that the following hold:

• •

For $x\in\{-1,1\}^{n+2}$, $wt(x)\in\{0,n+2\}$,

[TABLE]

• •

For $x\in\{-1,1\}^{n+2}$, $x_{n+1}=x_{n+2}=x_{1}$,

[TABLE]

• •

For $x\in\{-1,1\}^{n+2}$, $x_{n+1}\neq x_{n+2}$:

• –

If $c\cdot x\neq 0$,

[TABLE]

as it holds that $|\sum_{i=1}^{n}a_{i}x_{i}-y|+|\sum_{i=1}^{n}a_{i}x_{i}+y|=2\max(|\sum_{i=1}^{n}a_{i}x_{i}|,|y|)$ and $|\sum_{i=1}^{n}a_{i}x_{i}|\geq 1$.

• –

If $c\cdot x=0$, then

[TABLE]

• •

For $x\in\{-1,1\}^{n+2}$, $x_{n+1}=x_{n+2}$, $x_{1}\neq x_{n+1}$

[TABLE]

as similarly with the above case, we have that $|\sum_{i=1}^{n+2}a_{i}x_{i}-2y|+|\sum_{i=1}^{n+2}a_{i}x_{i}+2y|=2\max(|\sum_{i=1}^{n+2}a_{i}x_{i}|,|2y|)$ and $|\sum_{i=1}^{n+2}a_{i}x_{i}|\geq 1$.

Hence, we have:

[TABLE]

So, we get:

[TABLE]

The proof of Theorem 3 is now complete. ∎

We are now ready to prove Theorem 2.

Proof of Theorem 2.

We reduce from the Khintchine${}_{\mu_{p^{n+2}}}$ problem: given the vector $a=(a_{1},\dots,a_{n},a_{n+1}=-A/2,a_{n+2}=-A/2)\in\mathbb{Q}^{n+2}$, where $A=\sum_{i=1}^{n}a_{i}$ and $a_{i}>0$ for $0\leq i\leq n$, we want to compute $\max_{c\in\mathcal{C}_{p^{n+2}}}a\cdot c$. For any $c\in\mathcal{C}_{p^{n+2}}$, using our reformulation of semivalues (1), we have:

[TABLE]

where $f$ is a linear threshold function.

Reducing from a weight vector that has the sum of its coordinates equal to zero was essential for this step: the term $C_{f}^{p^{n+2}}$ vanishes and we end up with the term $\sum_{x\in\{-1,1\}^{n+2}}f(x)\mu_{p^{n+2}}(x)\sum_{i=1}^{n+2}a_{i}x_{i}$ that is upper bounded by $\sum_{x\in\{-1,1\}^{n+2}}\mu_{p^{n+2}}(x)|\sum_{i=1}^{n+2}a_{i}x_{i}|$.

This upper bound is tight, as we show below, which is crucial for our argument. If one were to reduce from a vector with sum different than [math] or include in the polytope only linear threshold functions with positive weights, it would not have been possible to obtain a tight upper bound.

Observe that

[TABLE]

for $c=(\widetilde{f}^{p^{n+2}}(1),\dots,\widetilde{f}^{p^{n+2}}(n+2))$, where $f(x)=\mathrm{sign}(\sum_{i=1}^{n+2}a_{i}x_{i}).$ So,

[TABLE]

That is, a solution to our instance of linear optimization over our polytope gives a solution to the initial instance of the Khintchine${}_{\mu_{p^{n+2}}}$ problem, and the proof of Theorem 2 is complete. ∎

Linear Optimization Using a Verification Oracle

We prove that the restricted verification problem, defined below, is computationally hard, where the input linear threshold functions are defined by weight vectors of the specific form described in Definition 4.

Name: SVR${}_{p^{n+2}}$-Verification Problem

Input: A vector $(c_{1},\dots,c_{n+2})\in\mathbb{Q}^{n+2}$ and a vector $w=(w_{1},\dots,w_{n+2})\in\mathbb{Q}^{n+2}$ such that $w_{n+1}=w_{n+2}=-\sum_{i=1}^{n}w_{i}/2$ and $w_{i}>0,i\in\{1,\dots,n\}$.

Question: Does it hold that $\widetilde{f}^{p^{n+2}}_{w,0}(i)=c_{i}$ for $i\in\{1,\dots,{n+2}\}$?

Theorem 4**.**

If $p^{n+2}$ is a reasonable probability vector, the SVR${}_{p^{n+2}}$-Verification problem is not in the $k$-th level of the polynomial hierarchy, unless #P is contained in the $(k+2)$-level.

The main idea behind the proof is that one can solve the linear optimization problem using a membership oracle of the polytope which can be obtained if we have an efficient algorithm for the restricted verification problem. Since the vertices of the polytope correspond to semivalues of linear threshold functions, if we have an efficient algorithm for the verification problem, then we can efficiently verify that a vector is a vertex of the polytope, and. using Caratheodory’s theorem, we can get a membership oracle. In this way, we obtain a contradiction, unless the polynomial hierarchy collapses: if the verification problem is in the polynomial hierarchy, then a #P-hard problem lies in the polynomial hierarchy.

The restricted membership problem for any probability vector $p^{n+2}$ is defined below:

Name: SVR${}_{p^{n+2}}$-Membership Problem

Input: A vector $c=(c_{1},\dots,c_{n+2})\in\mathbb{Q}^{n+2}$.

Question: Is $c$ in $\mathcal{C}_{p^{n+2}}$ ?

Proof of Theorem 4.

We first prove the following lemma that shows how an efficient oracle for the restricted verification problem can be used to obtain a membership oracle:

Lemma 1**.**

If the SVR${}_{p^{n+2}}$-Verification problem is in the $k$-th level of PH, then the SVR${}_{p^{n+2}}$-Membership problem is in the $(k+1)$-level.

Proof.

Assume that the SVR${}_{p^{n+2}}$-Verification problem is in the $k$-th level of PH. By Caratheodory’s theorem a point $c$ is in $\mathcal{C}_{p^{n+2}}$ iff it is a convex combination of at most $n+3$ vertices of $\mathcal{C}_{p^{n+2}}$, i.e., $c=\sum_{i=1}^{m}\lambda_{i}x^{(i)}$, where $x^{(i)}$ is a vertex, $\lambda_{i}\geq 0$ for $1\leq i\leq m\leq n+3$, and $\sum_{i=1}^{m}\lambda_{i}=1$. So, it can be certified that a given point $c$ is in $\mathcal{C}_{p^{n+2}}$ by finding the $m\leq n+3$ vertices $x^{(i)}$ and computing the $m$ scaling factors $\lambda_{i}$. Given the $x^{(i)}$, one can verify that $x^{(i)}$ is a vertex of $\mathcal{C}_{p^{n+2}}$ by finding a weight vector $w^{(i)}$ of the form described in Definition 4 such that $x^{(i)}$ is the $\widetilde{f}^{p^{n+2}}_{w^{(i)},0}$ vector, as the vertices of $\mathcal{C}_{p^{n+2}}$ correspond to linear threshold functions with weights of this specific form. So, if we are given the $x^{(i)}$ and the corresponding $w^{(i)}$, we can verify in polynomial time with a $k$-th level oracle that $x^{(i)}$ is the $\widetilde{f}^{p^{n+2}}_{w^{(i)},0}$ vector as we assumed that the SVR${}_{p}^{n+2}$-Verification problem is in the $k$-th level of PH. Thus, there is a polynomial-size certificate that can be checked in polynomial time with a $k$-th level oracle when $c$ is in $\mathcal{C}_{p^{n+2}}$: the $m\leq n+3$ vertices $x^{(i)}$, where each $x^{(i)}$ can be represented by poly($n$) bits by assumption; and the $m$ corresponding $w^{(i)}$ vectors, where each $w^{(i)}$ can be represented by poly($n$) bits, as every linear threshold function can be represented with weight $w=(w_{1},\dots,w_{n+2})$ such that each $w_{i}$ is an integer that satisfies $|w_{i}|\leq 2^{(O(n\log n))}$ [MTT61a]. Given the $x^{(i)}$ and the $w^{(i)}$, it can be verified in polynomial time with a $k$-th level oracle that the $x^{(i)}$ are vertices and then we can compute in polynomial time the $\lambda_{i}$ coefficients by solving the linear system $c=\sum_{i=1}^{m}\lambda_{i}x^{(i)}$. Thus, if the SVR${}_{p^{n+2}}$-Verification problem is in the $k$-th level of PH, the SVR${}_{p^{n+2}}$-Membership problem is in the $(k+1)$-level. ∎

As $\mathcal{C}_{p^{n+2}}$ has non-empty interior, if the SVR${}_{p^{n+2}}$-Membership problem is in the $(k+1)$-level of PH, then using the ellipsoid algorithm, we could solve the optimization problem using a polynomial number of membership-oracle calls (page 189, [Sch98]). Hence, we would have that the SV${}_{p^{n+2}}$-Optimization problem, which by Theorem 2 is $\#$P-hard, is in the $(k+2)$-level of PH. This completes the proof of Theorem 4. ∎

Hardness of the Verification Problem for Weighted Voting Games

One important issue is that the computational problems we have considered so far involve linear threshold functions some of whose weights can be negative. This seemed necessary to some extent for our arguments, as it is crucially exploited in the proof of Theorem 2.

We now show how to switch to weighted voting games (i.e., LTFs with non-negative weights), which was our initial goal. Using a bijection between the semivalues of a linear threshold function with weight vector $w=(w_{1},\ldots,w_{n})$ of the form described in Definition 4 and a linear threshold function with weight vector $w^{\prime}=(|w_{1}|,\ldots,|w_{n}|)$, we show the equivalence between the restricted verification problem and the verification problem for linear threshold functions defined by positive weights.

Name: SV${}_{p^{n}}$-Verification Problem

Input: A vector $(c_{1},\dots,c_{n})\in\mathbb{Q}^{n}$, a vector $w=(w_{1},\dots,w_{n})\in\mathbb{Q}_{+}^{n}$ and $\theta\in\mathbb{Q}$.

Question: Does it hold that $\widetilde{f}^{p^{n}}_{w,\theta}(i)=c_{i}$ for $i\in\{1,\dots,{n}\}$?

Theorem 5**.**

If the SV${}_{p^{n+2}}$-Verification problem is in the $k$-th level of PH, then the SVR${}_{p^{n+2}}$-Verification problem is in the $k$-th level of PH.

Proof.

We use the following lemma that shows how one can compute the semivalues of a linear threshold function with weight vector $w$ of the form described in Definition 4 given the semivalues of the linear threshold function with weight vector the absolute values of $w$, and vice-versa.

Lemma 2**.**

For a positive integer $n$, fix $a_{i}>0,1\leq i\leq n$, and $A=\sum_{i=1}^{n}a_{i}$. Consider the LTFs $f(x)=\mathrm{sign}(\sum_{i=1}^{n}a_{i}x_{i}-(A/2)x_{n+1}-(A/2)x_{n+2})$ and $g(x)=\mathrm{sign}(\sum_{i=1}^{n}a_{i}x_{i}+(A/2)x_{n+1}+(A/2)x_{n+2})$. Then, we have the following:

1. (i)

For $1\leq i\leq n$, it holds $\widetilde{g}^{p^{n+2}}(i)=\widetilde{f}^{p^{n+2}}(i)-2(p^{n+2}_{n+1}-p^{n+2}_{n-1})$, and 2. (ii)

For $n+1\leq i\leq n+2$, it holds $\widetilde{g}^{p^{n+2}}(i)=\widetilde{f}^{p^{n+2}}(i)+2\sum_{t=0}^{n-1}{\binom{n}{t}}(p^{n+2}_{t}+p^{n+2}_{t+1})$.

Proof.

For $i\in\{1,\dots,n\}$, from the definitions of semivalues and functions $f$, $g$ we get:

[TABLE]

[TABLE]

and

[TABLE]

As $f(x)=-1$ for any $x\neq\bf{1}$ with $x_{n+1}=x_{n+2}=1$, $f(x)=1$ for any $x$ with $x_{n+1}=x_{n+2}=-1$, $g(x)=1$ for any $x$ with $x_{n+1}=x_{n+2}=1$, and $g(x)=-1$ for any $x\neq(1,\dots,1,-1,-1)$ with $x_{n+1}=x_{n+2}=-1$, we get that

[TABLE]

For $i=n+1$, we can write:

[TABLE]

and

[TABLE]

We thus have

[TABLE]

In the same way, we get that

[TABLE]

This completes the proof of Lemma 2. ∎

Given an instance $(a_{1},\dots,a_{n},-A/2,-A/2)$, $(c_{1},\dots,c_{n+2})$ of the SVR${}_{p^{n+2}}$-Verification problem, we construct the following instance of SV${}_{p^{n+2}}$-Verification problem: $(a_{1},\dots,a_{n},+A/2,+A/2)$, $\theta=0$,

[TABLE]

By Lemma 2, we have that the SVR${}_{p^{n+2}}$-Verification instance is a “YES”-instance iff the SV${}_{p^{n+2}}$-Verification instance is a “YES”-instance. This completes the proof of Theorem 5. ∎

Verification Using Inverse Oracle

The final step of our proof is to show that the inverse problem for semivalues is at least as hard as the verification problem. While this is intuitively obvious, the proof requires the following non-trivial structural result: The semivalues of a weighted voting game characterize the game within the space of weighted voting games.

Theorem 6**.**

If the SV${}_{p^{n}}$-Inverse problem is in the $k$-th level of PH, then the SV${}_{p^{n}}$-Verification problem is in the $(k+1)$-level.

Proof.

The proof makes essential use of the following lemma that shows that if two LTFs with normalized weights and the same threshold have the same semivalues, then they are equal on all points $x$ that are given positive probability by the distribution $\mu^{\prime}_{p^{n}}$. This lemma is qualitatively similar to (and inspired by) Chow’s Theorem [Cho61], that shows that a linear threshold function is uniquely determined by its Chow parameters:

Lemma 3**.**

Let $f(x)=\mathrm{sign}(w\cdot x-\theta)$ and $g(x)=\mathrm{sign}(v\cdot x-\theta)$ where $\sum_{i=1}^{n}w_{i}=\sum_{i=1}^{n}v_{i}$. If $\widetilde{f}^{p^{n}}(i)=\widetilde{g}^{p^{n}}(i)$ for $i\in\{1,\dots,n\}$, then $f(x)=g(x)$ for all $x\in\{-1,1\}^{n}$ such that $p^{n}_{wt(x)}+p^{n}_{wt(x)-1}\neq 0$ and $|w\cdot x-\theta|+|v\cdot x-\theta|\neq 0$.

Proof.

By our assumption that $\widetilde{f}^{p^{n}}(i)=\widetilde{g}^{p^{n}}(i)$ for $i\in\{1,\dots,n\}$ it follows that:

[TABLE]

Recalling our reformulation of the semivalues (1), we equivalently have:

[TABLE]

Hence, for any $x\in\{-1,1\}^{n}$ such that $|w\cdot x-\theta|+|v\cdot x-\theta|\neq 0$ and $\mu^{\prime}_{p^{n}}(x)\neq 0$, we have that $f(x)=g(x)$. This completes the proof of Lemma 3. ∎

We are now ready to complete the proof. Given an SV${}_{p^{n}}$-Verification instance $a=(a_{1},\dots,a_{n})$, $\theta$, $(c_{1},\dots,c_{n})$, we create the following instance of the SV${}_{p^{n}}$-Inverse problem: $\theta^{\prime}=\theta/\sum_{i=1}^{n}{a_{i}}$, $(c_{1},\dots,c_{n})$. Then, if the SV${}_{p^{n}}$-Inverse instance is a “NO”-instance, we have a “NO”-instance of the SV${}_{p^{n}}$-Verification problem. If the SV${}_{p^{n}}$-Inverse output is a weight vector $w=(w_{1},\dots,w_{n})$, we can check with a co-NP oracle if the functions $f_{w,\theta^{\prime}}$ and $f_{a,\theta}$ have the same semivalues: By Lemma 3, they have the same semivalues iff there is no $x\in\{-1,1\}^{n}$ such that $\mu^{\prime}_{p^{n}}(x)>0$ and $f_{w,\theta^{\prime}}(x)\neq f_{a,\theta}(x)=f_{\frac{a}{\sum_{i=1}^{n}{a_{i}}},\frac{\theta}{\sum_{i=1}^{n}{a_{i}}}}(x)$. This completes the proof of Theorem 6. ∎

Theorem 1 now follows by combining Theorems 4, 5, and 6.

4 Conclusions

The inverse power index problem has received considerable attention in game theory and social choice, and the inverse Banzhaf index problem has been relevant in other fields as well, such as circuit complexity and computational learning. In this paper, we proved that the inverse semivalue problem, for reasonable probability distributions, is computationally intractable. As special cases, we deduce that the inverse Banzhaf index and inverse Shapley value problems are also intractable. A number of interesting open questions remain: Can we design efficient approximation algorithms for the inverse problem in the case of more general semivalues? Can we characterize the computational complexity of the inverse power index problem for power indices that do not belong in the semivalues class?

Acknowledgements.

We thank Shaddin Dughmi for his contributions to the early stages of this work. We are grateful to Kousha Etessami for numerous helpful discussions.