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. We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar "nice" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: @math Offline Hamming Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional Hamming space for @math , we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time @math or deterministic time @math . These also lead to faster MAX-SAT algorithms for sparse CNFs. @math Offline Approximate Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional @math or Euclidean space, we can find a @math -approximate nearest (or furthest) blue neighbor for each red point in randomized time near @math . @math SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for @math circuits with a subquadratic number of linear threshold gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic @math time. This also implies new circuit lower bounds for threshold circuits. We also give a randomized @math -time SAT algorithm for subexponential-size @math circuits, where the top @math gate and middle @math gates have @math fan-in. Weighted voting games provide a simple model of decision-making in human societies and multi-agent systems. Such games are described by a set of players, a list of [email protected]? weights, and a quota; a coalition of the players is said to be winning if the total weight of its members meets or exceeds the quota. The power of a player in a weighted voting game is traditionally identified with her Shapley-Shubik index or her Banzhaf index, two classic power measures that reflect the [email protected]?s marginal contribution under different coalition formation scenarios. In this paper, we investigate by how much one can change a [email protected]?s power, as measured by these indices, by modifying the quota. We give tight bounds on the changes in the individual [email protected]?s power that can result from a change in quota. We then describe an efficient algorithm for determining whether there is a value of the quota that makes a given player a dummy, i.e., reduces her power (as measured by both indices) to 0. We also study how the choice of quota can affect the relative power of the players. Finally, we investigate scenarios where [email protected]?s choice in setting the quota is constrained. We show that optimally choosing between two values of the quota is complete for the complexity class PP, which is believed to be significantly more powerful than NP. On the other hand, we empirically demonstrate that even small changes in quota can have a significant effect on a [email protected]?s power. A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243–260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation. The paper investigates the complexity of depth-two circuits with threshold gates and consisting of two parts. First, we develop a method for deriving a lower bound on the size of depth two circuits with a threshold gate at the top and a certain type of gates at the bottom. We apply the method for circuits with symmetric gates at the bottom that compute the “inner product mod 2”, and obtain a lower bound of 1.3638n. Although our lower bound is slightly weaker than the best known lower bound of Ω(2n 2 n), which was recently proved by [5,6], our method has unique features: A lower bound is obtained by solving a certain linear program, and solving larger linear programs yield higher lower bounds. We also discuss the generalization of the proposed method. Second, we develop a simplified simulation of a depth-one threshold circuit with unbounded weights by a depth-two threshold circuit with small weights. Precisely, we give an explicit construction of depth-two circuits with small weights consist of On5 gates that compute an arbitrary threshold function. We also give the construction of such circuits with O(n3 log n) gates computing the COMPARISON and CARRY functions, and that with O(n4 log n) gates computing the ADDITION function. These improve the previously known constructions on its size and simplicity. Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index. In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these algorithms. Finally, we examine related forms of manipulative behavior, such as annexation, where a player subsumes other players, or merging, where several players unite into one. We characterize the computational complexity of such manipulations and provide limits on their effects. For the Banzhaf index, we describe a new paradox, which we term the Annexation Non-monotonicity Paradox. This article initiates a rigorous theoretical analysis of the computational power of circuits that employ modules for computing winner-take-all. Computational models that involve competitive stages have so far been neglected in computational complexity theory, although they are widely used in computational brain models, artificial neural networks, and analog VLSI. Our theoretical analysis shows that winner-take-all is a surprisingly powerful computational module in comparison with threshold gates (also referred to as McCulloch-Pitts neurons) and sigmoidal gates. We prove an optimal quadratic lower bound for computing winner-take-all in any feedforward circuit consisting of threshold gates. In addition we show that arbitrary continuous functions can be approximated by circuits employing a single soft winner-take-all gate as their only nonlinear operation. Our theoretical analysis also provides answers to two basic questions raised by neurophysiologists in view of the well-known asymmetry between excitatory and inhibitory connections in cortical circuits: how much computational power of neural networks is lost if only positive weights are employed in weighted sums and how much adaptive capability is lost if only the positive weights are subject to plasticity. In many multiagent settings, situations arise in which agents must collectively make decisions while not every agent is supposed to have an equal amount of influence in the outcome of such a decision. Weighted voting games are often used to deal with these situations. The amount of influence that an agent has in a weighted voting game can be measured by means of various power indices. This paper studies the problem of finding a weighted voting game in which the distribution of the influence among the agents is as close as possible to a given target value. We propose a method to exactly solve this problem. This method relies on a new efficient procedure for enumerating weighted voting games of a fixed number of agents. The enumeration algorithm we propose works by exploiting the properties of a specific partial order over the class of weighted voting games. The algorithm enumerates weighted voting games of a fixed number of agents in time exponential in the number of agents, and polynomial in the number of games output. As a consequence we obtain an exact anytime algorithm for designing weighted voting games.