We study the inverse power index problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distribution. Our goal is to find algorithms that solve this problem exactly. Thereto, we study various subclasses of simple games, and their associated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions (also known as shift-maximal losing coalitions) of a game from its list of roof coalitions (also known as shift-minimal winning coalitions), and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the synthesis problem, we then improve on this naive algorithm, and we obtain an enumeration algorithm that runs in quadratic exponential time (that is, O(2^(n^2) p(n)) for a polynomial p). Moreover, we show that this algorithm runs in output-polynomial time, making it the best possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime algorithm for the inverse power index problem that runs in exponential time. This algorithm is straightforward and general: it computes the error for each game enumerated, and outputs the game that minimizes this error. By the genericity of our approach, our algorithm can be used to find a weighted voting game that optimizes any exponential time computable function. We implement our algorithm for the case of the normalized Banzhaf index, and we perform experiments in order to study performance and error convergence.
Closely related to our work are two papers that deal with the Chow parameters problem @cite_65 @cite_50 . The results in their paper are stated in terms of boolean function theory and learning theory, but when translated to our setting, these papers can be seen to deal with approximation algorithms for a type of value that can be considered a power index: The Chow parameters of a given player in a given game is defined to be the total number of winning coalitions that the players is in. The authors present in these papers, as a main result, a polynomial time approximation scheme for computing the Chow parameters of a weighted voting game.
@cite_65 @cite_50
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The Chow parameters of a Boolean function f: -1,1 n -> -1,1 are its n+1 degree-0 and degree-1 Fourier coefficients. It has been known since 1961 [Cho61, Tan61] that the (exact values of the) Chow parameters of any linear threshold function f uniquely specify f within the space of all Boolean functions, but until recently [OS11] nothing was known about efficient algorithms for reconstructing f (exactly or approximately) from exact or approximate values of its Chow parameters. We refer to this reconstruction problem as the Chow Parameters Problem. Our main result is a new algorithm for the Chow Parameters Problem which, given (sufficiently accurate approximations to) the Chow parameters of any linear threshold function f, runs in time O(n2)• (1 e)O(log2(1 e)) and with high probability outputs a representation of an LTF f' that is e-close to f. The only previous algorithm [OS11] had running time poly(n) • 22 O(1 e2). As a byproduct of our approach, we show that for any linear threshold function f over -1,1 n, there is a linear threshold function f' which is e-close to f and has all weights that are integers at most √n • (1 e)O(log2(1 e)). This significantly improves the best previous result of [Serv09] which gave a poly(n) • 2O(1 e2 3) weight bound, and is close to the known lower bound of max √n, (1 e)Ω(log log (1 e)) [Gol06,Serv07]. Our techniques also yield improved algorithms for related problems in learning theory. In addition to being significantly stronger than previous work, our results are obtained using conceptually simpler proofs. The two main ingredients underlying our results are (1) a new structural result showing that for f any linear threshold function and g any bounded function, if the Chow parameters of f are close to the Chow parameters of g then f is close to g; (2) a new boosting-like algorithm that given approximations to the Chow parameters of a linear threshold function outputs a bounded function whose Chow parameters are close to those of f. In [Proceedings of the Second Symposium on Switching Circuit Theory and Logical Design (FOCS), 1961, pp. 34-38], Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow parameters. Providing an algorithmic version of Chow's theorem—i.e., efficiently constructing a representation of a threshold function given its Chow parameters—has remained open ever since. This problem has received significant study in the fields of circuit complexity, game theory and the design of voting systems, and learning theory. In this paper we effectively solve the problem, giving a randomized polynomial-time approximation scheme with the following behavior: Given the Chow parameters of a Boolean threshold function @math over @math bits and any constant @math , the algorithm runs in time @math and with high probability outputs a representation of a threshold function @math which is @math -close to @math . Along the way we prove several new results of independent interest about Boolean threshold functions. In addition to various structural results, these include @math -time learning algorithms for threshold functions under the uniform distribution in the following models: (i) the restricted focus of attention model, answering an open question of ; (ii) an agnostic-type model. This contrasts with recent results of Guruswami and Raghavendra who show NP-hardness for the problem under general distributions; (iii) the PAC model, with constant @math . Our @math -time algorithm substantially improves on the previous best known running time and nearly matches the @math bits of training data that any successful learning algorithm must use. Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices, which reflect how much "real power" a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate power indices such as the Banzhaf power index and the Shapley---Shubik power index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game's value for several sample coalitions. We show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating power indices. We also provide empirical results regarding our method, and show that it typically achieves much better accuracy and confidence than those required. Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices, which reflect how much "real power" a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate power indices such as the Banzhaf power index and the Shapley-Shubik power index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game's value for several sample coalitions. We also show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating power indices.