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.
Threshold functions @cite_5 @cite_29 are of fundamental research interest in voting games, circuit complexity and neural networks. The problem of realizing Boolean threshold functions by neural networks has been extensively studied @cite_51 @cite_66 @cite_34 , where upper and lower bounds are derived on the synaptic weights for such realization. The enumeration of threshold functions is closely related to the enumeration of weighted voting games (as threshold functions are essentially weighted voting games where negative weights are allowed.). The enumeration of threshold functions up to six variables has been done in @cite_52 . Subsequently, in @cite_39 @cite_0 , all threshold functions of respectively seven and eight variables were enumerated. Krohn and Sudh "olter @cite_23 enumerated the canonical weighted voting games up to eight players, as well as the class of canonical linear games. Kurz @cite_14 was the first to enumerate all nine player canonical weighted voting games, and Freixas and Molinero @cite_6 were the first to enumerate all nine player canonical linear games.
@cite_14 @cite_29 @cite_52 @cite_6 @cite_39 @cite_0 @cite_23 @cite_5 @cite_34 @cite_51 @cite_66
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Threshold functions and related operators are widely used as basic elements of adaptive and associative networks [Nakano 72, Amari 72, Hopfield 82]. There exist numerous learning rules for finding a set of weights to achieve a particular correspondence between input-output pairs. But early works in the field have shown that the number of threshold functions (or linearly separable functions) in N binary variables is small compared to the number of all possible boolean mappings in N variables, especially if N is large. This problem is one of the main limitations of most neural networks models where the state is fully specified by the environment during learning: they can only learn linearly separable functions of their inputs. Moreover, a learning procedure which requires the outside world to specify the state of every neuron during the learning session can hardly be considered as a general learning rule because in real-world conditions, only a partial information on the “ideal” network state for each task is available from the environment. It is possible to use a set of so-called “hidden units” [Hinton,Sejnowski,Ackley. 84], without direct interaction with the environment, which can compute intermediate predicates. Unfortunately, the global response depends on the output of a particular hidden unit in a highly non-linear way, moreover the nature of this dependence is influenced by the states of the other cells. The number of threshold functions of eight variables is counted by ILLIAC II, the computer of the University of Illinois. Sets of optimum weights of majority elements realizing these functions also are investigated. Actually, canonical positive self-dual threshold functions of nine variables are investigated instead of directly investigating threshold functions of eight variables because it is easier to deal with them. The number and optimum weights of threshold functions of eight variables are easily obtained from these functions of nine variables and their realization. In this paper we give structural characterizations of disjunctive and conjunctive hierarchical simple games by characterizing them as complete games with a unique shift-maximal losing coalition, and a unique shift-minimal winning coalition, respectively. We prove canonical representation theorems for both types of hierarchical games and establish duality between them. We characterize the disjunctive and conjunctive hierarchical games that are weighted majority games. This paper was inspired by (2008) and Farras and Padro (2010) characterizations of ideal weighted threshold access structures of secret sharing schemes. Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function @math . However, its naive encoding needs @math bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using @math weights and one threshold. For heterogeneous agents, one can represent @math as an intersection of @math threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring @math and provided a construction guaranteeing @math . The magnitude of the worst-case situation was thought to be determined by in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number @math for a subclass of voting systems. As an application, we give a construction for @math , i.e., there is no gain from a representation complexity point of view.